LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


UtM/Sr^l 

" 

Class 


ART  OF  COMPUTATION 


AND 


:R/U:L,:E 


/  FOR 


EQUATION  OF  PAYMENTS. 


ONE    DOLLAR. 


-        OF  THE 

UNIVERSITY 

OF 


HOWARD'S 


JIRT  OF  COMPUTATION 


OF  THE 

UNIVERSITY 

OF 


AND 

GOLDEN  RULE 

FOR 


EQUATION  OF  PAYMENTS 

FOR 

SCHOOLS,  BUSINESS  COLLEGES 

AND 

SELF-CULTURE. 

A  NEW,  CONCISE  AND  COMPREHENSIVE 


TEACHER  AND  MANUAL 
OP 


BUSINESS  ARITHMETIC. 


;»Y  TKK  AUTHOR  OP  THK  CALIFORNIA  CAJLCOI*ATOK> 

C.    FKUSHER    HOWARD, 

SAN  FRANCISCO. 

1880. 


Entered  according  to  Act  of  Congress,  in  the  year  of  our 
Lord  1879,  by  C.  Frusher  Howard,  in  the  office  of  the  Libra- 
rian of  Congress,  Washington,  D.  C. 

ALL  EIGHTS  SECURED. 


The  Author  specially  cautions  all  Book  Pirates  that 
his  sole  rights  and  title  to  the  following  original  Rules  and 
Tables  are  legally  secured,  and  will  be  maintained  against 
all  infringements. 
HOWARD'S  Golden  Rule  for  Equation  of  Payments. 

Averaging  Accounts. 
"  "          "  Partial  Payments. 

Computing  Interest  on  a  Basis  of  one  per  cent. 

"        by  dividing  the  year  by  the  rate* 
"  "        Bank  of  England  Rule. 

Compound  Interest. 

Squaring  numbers  by  their  base  and  difference. 
California  Calendar  for  thirty  centuries. 
The  original  Tables  and  their  arrangement. 


"HE 

MIVERSITY 

PREFACE. 


?HE  ability  to  make  business  calculations  with 
ease,  accuracy  and  rapidity,  is  an  all-important 
acquisition  to  every  class  of  the  community. 
The  methods  of  Arithmetic  hitherto  taught  have 
been  so  abstruse  and  difficult  as  to  deter  all,  but  a 
small  per  centage,  from  giving  the  weary  months 
and  years  of  time,  labor  and  study  necessary  to 
master  its  mysteries.  WONDERFUL  and  STARTLING 
discoveries  have  RECENTLY  been  made  and  em- 
bodied in  the  following  rules,  simplify  ing- and  short- 
ening all  the  operations  of  numbers,  so  as  to  make 
RAPID  CALCULATION  easy  to  all; 

The  rules  taught  in  Schools  are  needlessly  weight- 
ed with  superfluous  elements,  that  only  serve  to  en- 
cumber the  operations,  and  distract,  and  confuse  the 
learner ;  the  Rules  here  taught  avoid  all  this,  and 
by  an*easily  learned,  simple,  and  natural  arrangement 
lead  directly  to  the  required  answer. 

They  are  especially  adapted  to  that  large  class  of 
persons  who  find  it  difficult,  or  impossible,  mentally 
to  grasp,  and  retain  complex  numbers  ;  such  persons 
will  find  in  this  book 

"A  Complete  Teacher  of  Business  Arithmetic" 
all  the  examples  being  worked  out,  and  explained  so 
as  to  be  readily  understood,  transforming  the  drudg- 
ery of  calculation,  into  a  pleasing  pastime,  and  qual- 
ifying persons  of  ordinary  intellect,  to  surpass  the 
performances  of  the  "Lightning  Calculators"  who 
have  astonished  mankind. 

153721 


PREFACE. 

The  success  achieved  by  the  CALIFORNIA  CALCULATOR 
encourages  the  hope  that  the  ART  OF  COMPUTATION  will 
soon  be  in  every  school,  making  ALL  the  Boys  "quick  at 
figures"  The  more  gifted  and  ambitious  will  become 
expert  Mathematicians  with  greater  facility  if  they  are 
FIRST  good  calculators. 

A  knowledge  of  the  SCIENCE  of  numbers  is  an  invalu- 
able acquisition  to  those  who  are  capable  of  acquiring  it; 
to  do  this  not  more  than  one  person  in  a  hundred  has 
either  the  time,  necessity,  or  mental  capacity. 

To  Accountants,  Brokers,  Farmers,  Traders  and  person? 
engaged  in  the  ruder  mechanical  pursuits,  a  knowledge 
of  the  SCIENCE  OF  NUMBERS  is  of  minor  importance  ;  SKILL 
in  the  ART  OF  COMPUTATION  is  absolutely  indispensable; 
the  business  of  this  Book  is  by  new,  original  and  easily  ac? 
quired  methods  to  teach  that  ART,  in  accord  with,  yet  dis- 
tinct from  the  SCIENCE. 

As  a  SCHOOL  BOOK,  its  aim  is  to  make  the  learner  a  GOOD 
CALCULATOR  with  the  greatest  possible  economy  of  time 
and  study;  its  preeminence  consists  in  the  brevity  and 
clearness  of  the  rules ;  by  their  use,  Interest  and  other 
calculations  may  be  made,  easier  than  they  can  be  copied 
from  ordinary  Tables. 

The  REFERENCE  TABLES  are  very  comprehensive  and 
their  arrangement  simple  and  original. 

The  miscellaneous  section  is  unique ;  it  embraces  almost 
every  variety  of  BUSINESS  CALCULATION,  the  work  of  find- 
ing the  answer  to  each  question  is  so  expressed  that  it 
constitutes  a  formula  for  all  similar  examples. 

One  Reviewer  of  these  Rules  and  Tables  says: 

"  Students,  Teachers  and  Business  Men  can  no  more 
afford  to  be  without  them  than  they  can  afford  to  travel 
by  OX-TEAMS,  now  the  RAILWAY  spans  the  Continent." 


TABLE  OF  CONTENTS: 


Addition, 16 

Aliquot  Parts, 32 

British  Money,  70 

Cancellation,  40 

Compound  Interest, 65 

Definitions  and  Signs, 7 

Decimals, 35 

Division, 27 

Discount, 66 

Exchange,  69 

Fractions, 29 

Gold  and  Silver, 100,  101,  102,  111 

Multiplication, 19 

Measuring  Land, 47,  104 

"         Timber, 56,105 

Marking  Goods, 92 

Miscellaneous,  114 

Notation,  14 

Numeration, 15 

Percentage,  72 

Subtraction, 19 

Rapid  Rules  for  Farmers, 45 

"      Reckoning  for  Mechanics,  51 

"      Method  for  Squaring  Numbers, 24 

"      Rules  for  COMPUTING  INTEREST,  59 

Rules  for  Money  and  Bullion  Brokers,...  100,  111 

Tables  for  Business  Reference, 93 

Tables  of  Standard  Weights  and  Measures, 10S 


6  TABLE    OF    CONTENTS. 

PAGE 

Proportion 39 

Rapid  Bale  for  reckoning  the  cost  of  Hay 46 

Subtraction  of  Fractions  31 

Square  and  Cube  Root 87 

Stocks  and  Bonds 74 

To  find  the  Greatest  Common  Factor  or  Divisor..  30 
To  find  the  value  of  Grain  per  Cental,  or  Bushel, 

the  price  of  either  being  given 45 

To  Measure  Grain 46 

To  Measure  Land  without  Instruments 47 

To  lay  oif  a  Square  Corner  51 

To  Measure  Grindstones 52 

To  Measure  Superfices  and  Solids  52 

„  Bricklayers'  Work  57 

„  Plasterers'  Work 58 

Painters'  Work 58 

„  Gangers'  Work 57 

To  find  the  Difference  of  Time  between  any  two 

Dates ..110,  83 

Howard's  New  Rule  for  Interest  on  a  basis  of  1°/0  59 

Howard's  California  Calendar  for  Thirty  Centuries  86 
To  find  the  value  of  Gold  or  Currency,  the  price 

of  either  being  given 69 

The  Number  Nine  91 

Percentage <  .  72 

Partial  Payments .' . .  82 

Averaging  Accounts 81 

Cash  Balances  •» 83 

Golden  Bule  for  Equation  of  Payments 75 


HOWARD'S 


ART  OF  COMPUTATION. 


DEFINITIONS  AND  SIGNS. 

ARITHMETIC  is  the  science  of  numbers,  and  the 
art  of  computing  by  figures. 

ABSTRACT  NUMBER. —  An  abstract  number  is  a 
number  used  without  reference  to  any  particular 
object,  as  9,  745,  9764. 

ADDITION,  the  act  of  adding,  opposed  to  subtrac- 
tion. 

AMOUNT. — The  sum  of  principal  and  interest. 

ALIQUOT. — An  aliquot  part  of  a  number  is  such  a 
part  as  will  exactly  divide  that  number. 

AREA,  the  surface  included  within  any  given 
lines. 

ARITHMETICAL  SIGNS  are  characters  indicating 
operations  to  be  performed,  and  are  indispensable 
for  briefly  and  clearly  stating  a  problem : 

-j-,pfa£S,  and  more,  signifying  addition; 

— ,  minus,  less,  signifying  subtraction; 


8  HOWARD'S  ART  OF  COMPUTATION. 

X,  multiplied  by,  as  2  X  2  =  4  ; 

~  or  :  divided  by,  as  6  -f-  3  =  2,  or  6  :  3  =  2, 
orf  =  2; 

=,  equality,  or  is  equal  to,  as  6  -f  2  X  2  =  16, 
and  is  read  thus,  "  6  plus  2,  multiplied  by  2, 
equals  16  "; 

,  or  (     )  &c.T  the  vinculum ;  used  to  shew  that 

all  the  numbers  united  by  it  are  to  be  considered  as 
one ;  thus,  6x4  +  3x2  +  1  means  the  product  of 
6  X  4  is  to  be  added  to  the  product  of  3  X  2,  and  the 
sum  of  the  products  to  be  added  to  1. 

V  9,  sign  of  the  square  root,  read  "the  square 
root  of  9  " ; 

42,  sign  of  the  square,  read  "the  square  of  4"; 

^8,  the  cube  root  of  8.     83,  the  cube  of  8. 

AN  ANGLE  is  the  corner  formed  by  two  lines 
where  they  meet. 

BASE,  the  lower,  or  side  upon  which  a  figure 
stands;  the  foundation  of  a  calculation. 

CONCRETE  NUMBER,  used  with  reference  to  some 
particular  object  or  quantity,  as  640  acres,  500 
dollars. 

CIRCLE,  a  plane  figure  comprehended  by  a  single 
curved  line,  called  its  circumference,  every  part  of 
which  is  equidistant  from  its  center. 

CIRCUMFERENCE,  the  line  that  goes  around  a 
circle  or  sphere. 

CYLINDER,  a  body  bounded  by  a  uniformly 
curved  surface,  its  ends  being  equal  and  parallel 
circles. 


DEFINITIONS  AND  SIGNS.  9 

CUBE,  a  solid  body  with  six  equal  square  sides. 
A  product  formed  by  multiplying  an}'  number 
twice  by  itself,  as4x  4  X  4  =  6  4,  the  cube  of  4. 

CUBE  ROOT  is  the  number  or  quantity  which 
twice  multiplied  into  itself  produces  the  number  of 
which  it  is  the  root,  thus  4  is  the  cube  root  of  64. 

CURRENCY,  the  current  medium  of  trade  author- 
ized by  government. 

DIVISION  determines  how  many  times  any  one 
number  is  contained  in  another. 

DISCOUNT,  the  sum  deducted  from  an  account,, 
note,  or  bill  of  exchange,  usually  at  some  rate  per 
cent. 

DENOMINATOR,  the  number  placed  below  the  line 
in  fractions,  thus,  in  j-  (seven-eights)  8  is  the  de- 
nominator. 

DECIMAL,  a  tenth ;  a  fraction  having  some  power 
of  10  for  its  denominator. 

DECIMAL  CURRENCY  is  a  currency  whose  denom- 
inations increase  or  decrease  in  a  ten-fold  ratio. 

DIVIDEND,  the  number  to  be  divided. 

DIVISOR,  the  number  by  which  the  dividend  is  to- 
be  divided.  A  common  divisor,  is  a  number  that 
will  divide  two  or  more  numbers  without  a  remain- 
der. 

DIAMETER,  a  right  line  passing  through  any 
object. 

DUODECIMALS  are  the  divisions  and  subdivisions. 


10  HOWARD'S  ART  OF  COMPUTATION. 

of  a  unit,  resulting  from  continually  dividing  by 
12,  as  1,  TV,  Ti¥,  TTV¥,  etc. 

EXCHANGE,  the  receiving  or  paying  of  money  in 
one  place  for  its  value  in  another,  by  order, 
•draft,  or  bill  of  exchange. 

FRACTION,  part  or  parts  of  a  whole  number  or 
unit,  thus  J,  three-fourths,  |,  one-fifth. 

An  improper  fraction  is  a  fraction  whose  numera- 
tor exceeds  its  denominator. 

FACTORS,  numbers,  from  the  multiplication  of 
which  proceeds  the  product ;  thus,  3  and  4  are  the 
factors  of  12. 

FIGURE — A  figure  is  a  written  sign  representing 
&  number. 

INTEGER — An  integer  is  a  whole  number  or  sum. 

INTEREST,  the  price  or  sum  per  cent,  derived  from 
the  use  of  money  lent.  Simple  interest  is  that  which 
arises  from  the  principal  sum  only.  Compound 
interest  is  that  which  arises  from  the  principal  and 
interest  added — interest  on  interest. 

MATHEMATICS,  the  science  of  quantities. 

MULTIPLICATION,  adding  to  zero  any  given  num- 
ber as  many  times  as  there  are  units  in  the  mul- 
tiplier. 

MULTIPLIER,  the  number  that  multiplies ;  the 
multiplier  must  be  an  abstract  number. 

MULTIPLICAND,  the  number  multiplied. 

MENSURATION  is  the  art  of  measuring  lengths, 
eurfaces,  and  solids. 


DEFINITIONS  AND  SIGNS.  11 

MULTIPLE,  a  quantity  which  contains  another  a 
certain  number  of  times  without  a  remainder.  A 
common  multiple  of  two  or  more  numbers  contains 
each  of  them  a  certain  number  of  times,  exactly. 
The  least  common  multiple  is  the  least  number  that 
will  do  this;  12  is  the  least  common  multiple  of  3 
and  4. 

NUMBER,  a  number  is  a  unit,  or  a  collection  of 
units.  A  prime  number  is  one  that  cannot  be  re- 
solved, or  separated  into  two  or  more  integral 
factors. 

NOTATION,  writing  numbers. 

NUMERATION,  reading  numbers. 

NUMERATOR,  the  number  placed  above  the  line,  in 
fractions;  thus,  j;  (five-ninths),  five  is  the  numerator. 

POWER — A  power  is  the  product  arising  from 
multiplying  a  number  by  itself,  or  repeating  it 
several  times  as  a  factor;  thus,  3x3x3,  the 
product,  27,  is  a  power  of  3.  The  exponent  of  a 
power  is  the  number  denoting  how  many  times  the 
factor  is  repeated  to  produce  the  power,  and  is 
written  thus:  21,  23,  23. 

21  =  21  =  2,  the  first  power  of  2. 
2x2=22=4,  the  second  power  of  2. 
2  X  2  X  2  =  23  =  8,  the  third  power  of  2. 

PRINCIPAL,  the  sum  lent  on  interest,  or  invested. 

PER  CENT.,  from  per  centum,  signifying  by  the 
hundred ;  hence,  1  per  cent,  of  anything  is  one-hun- 
dredth part  of  it,  2  per  cent,  is  one-fiftieth,  etc. 


12          HOWARD'S  ART  OF  COMPUTATION. 

QUADRANGLE,  the  name  of  a  figure  with  four 
sides. 

QUANTITY  is  anything  that  can  be  increased, 
diminished,  or  measured. 

RATIO,  the  quotient  of  one  number  divided  by 
another. 

RECIPROCAL  is  a  unit  divided  by  any  number. 
The  reciprocal  of  any  number  or  fraction,  is  that 
number  or  fraction  inverted ;  thus  the  reciprocal  of 
£  is  i,  off  is  ,f  of  3£  is  T3T. 

RATE  PER  CENT.,  the  rate  per  hundred. 

• 

RULE — A  rule  is  the  prescribed  method  of  per- 
forming an  operation. 

RADIUS,  half  the  diameter  of  a  circle.  A  right 
line  passing  from  the  center  to  the  circumference. 

SUBTRACTION  is  the  process  of  finding  the  differ- 
ence of  two  numbers  by  taking  one  number  called  the 
subtrahend  from  another  number  called  the  minuend. 

SURFACE  or  SUPERFICES,  the  exterior  part  of 
anything  that  has  length  and  breadth. 

SUPPLEMENT,  the  difference  of  a  number  and  some 
particular  number  below  it;  thus  13,  taking  10  as 
:he  base,  the  supplement  is  3,  because  the  difference 
of  13  and  10  is  3. 

SQUARE,  a  figure  having  four  equal  sides,  and 
four  right  angles.  The  product  of  a  number 


DEFINITIONS    AND    SIGNS.  13 

multiplied  by  itself;  thus  16  is  the  square  of  4. 
4  X  4  —  16. 

SQUARE  ROOT  is  the  number  which  multiplied 
into  itself,  produces  tlie  number  of  which  it  is  the 
root.  4  is  the  root  of  16;  4  x  4  =  16. 

SPECIE,  coin. 

SCALE — A  scale  is  a  series  of  numbers  regularly 
ascending  or  descending. 

A  SOLID  or  BODY  has  length,  breadth  and  thick- 
ness. 

SPHERE,  a  body  in  which  every  part  of  the  sur- 
face is  equally  distant  from  the  center. 

TRIANGLE,  a  figure  with  three  sides. 

TERM — The  terms  of  a  fraction  are  numerator 
and  denominator  taken  together. 

UNIT — A  unit  is  one  thing. 

VERTEX,  the  top  of  a  pyramid  or  cone. 

ZERO,  a  cipher,  or  nothing. 

In  arithmetic,  the  answer  in  each  operation  has 
a  distinctive  name.  In  addition  it  is  called  the 
sum;  in  subtraction,  difference  or  remainder;  in 
multiplication,  the  product;  in  division,  the  quo- 
tient. 


14  HOWARD'S  ART  OF  COMPUTATION. 


NOTATION. 


All  numbers  are  represented  by  the  ten  follow- 
ing figures  : 

s    £  1    g    £  js  I  I  1  I 

O+j+3«Scacea5<i>flo 
1,      2,      3,      4,      5,      6,      7,      8,      9,      0. 

To  establish  their  significance  clearly  in  the 
mind  of  the  pupil  it  will  be  of  great  advantage 
occasionally  to  write  and  read  them  in  the  follow- 
ing manner  : 


•  i 

1    5°   ?i   ^   si   so  ice,  co  §  GO  "§   so   cr>  02  cn 

o  o  o  g  o>  s  o>  o  o>  i!|  o  u  o  *3v  o>  S  3>  S 

a  jt  fl  Ja  fl  S-g  ^  «  .8  a  S  a.SP  S  .9  ^  o  S 


1234567 
TTTT 


T         T         T 

The  different  values  which  the  same  figures  have, 
are  called  simple  and  local  values. 

The  -simple  value  of  a  figure  is  the  value  it 
expresses  when  it  stands  alone,  or  in  the  right 
hand  place. 

The  local  value  of  a  figure  is  the  increased  value 
which  it  expresses  by  having  other  figures  placed 
on  its  right. 

Ten  is  expressed  by  combining  one  and  cipher, 
thus,  10;  two  and  cipher  combined  make  twenty, 
thus,  20,  etc.  A  hundred  is  expressed  by  combin- 
ing the  one  and  two  ciphers,  thus,  100;  two 


NUMERATION.  ]  5 

hundred  thus,  200,  etc.  Ten  ones  make  a  ten ;  ten 
tens  make  a  hundred ;  ten  hundreds  make  one  thou- 
sand; that  is,  numbers  increase  from  right  to  left 
in  a  ten-fold  ratio.  Each  removal  of  a  figure  one 
place  to  the  left  increases  its  value  ten  times. 


NUMEEATION. 


!  1  I 


.         .9  .2        s    s    «    §    a 

^3'!.^Z2  GoC5ScQ 

1  j  1  1  1  I  1  1  I  1  I  1 

HQPQ!z;Occc/2O>O>HW^H£> 
121,227,196,497,821,415,716,219,304,196,218,316,415,207,126. 

To  read  numbers  expressed  by  figures  :  Point 
them  off  into  periods  of  three  figures  each,  com- 
mencing at  the  right  hand  ;  then,  beginning  at 
left  hand,  read  the  figures  of  each  period  in  the 
same  manner  as  those  of  the  right  hand  period 
are  read,  and  at  the  end  of  each  period  pronounce 
its  name;  thus,  121  tredecillions,  227  duodecil- 
lions,  196  undecillions,  497  decillions,  321  nonil- 
lions,  415  octillions,  716  septillions,  219  sextillions, 
304  quintillions,  196  quadrillions,  218  trillions,  31B 
billions,  415  millions,  207  thousands,  126. 


16  HOWARD'S  ART  OF  COMPUTATION. 


ADDITION. 

Variotas  suggestions  have  been  made  referring  to 
improved  methods  of  addition.  In  nearly  every 
ease  the  proposed  improvement  has  been  more 
fanciful  than  real.  In  practice,  I  have  found  no 
better  or  quicker  method  than  the  following: 

3746 
8743 
6978 
1256 
3021 


23744 

Commence  at  the  bottom  of  the  right  hand  col- 
umn; add  thus,  7,  15,  18,  24;  set  down  the  4  in 
unit's  place,  and  carry  the  two  tens  to  the  second 
column;  then  add  thus,  4,  9,  1C,  24;  set  down  the 
4  in  ten's  place,  and  carry  the  two  hundreds  to  the 
third  column,  and  so  on  to  the  end.  Never  add 
in  this  manner:  1  and  6  are  seven,  and  8  are  15, 
and  3  are  18,  and  6  are  24.  It  is  just  as  easy  to 
name  the  sum  at  once,  omitting  the  name  of  each 
separate  figure,  and  saves  two  thirds  of  time  and 
labor. 

Book-keepers  and  others  who  have  long  columns 
of  figures  to  add  will  find  the  following  methods 
and  suggestions  acceptable. 


ADDITION.  17 

Rule  of  addition  for  two  columns  at  once  :  first 
practice  adding  two  columns  of  two  figures  each,  until 
you  are  able  to  grasp  at  a  glance,  and  pronounce  their 
sum. 

23    15    32    38    87    56 
14    33'   44    57    41    78 


37        48        76         95128134 
Add  from  the  left,  and  say  three  seven,  four  eight, 
twelve  eicfht-,  &c.,  &c.,  instead  of  thirty-seven,  forty-eight, 
one  hundred  and  twenty-eight,  &c.,  &c. ;   this  habit  is 
readily  acquired  and  saves  half  the  time. 

When  you  can  instantly,  at  sight,  name  the  sum  of 
two  pairs  of  figures,  practice  with  gradually  increasing 
columns  of  pairs,  then  take  examples  consisting  of  two 
or  more  columns  of  pairs. 


36 

2147 

41 

3472 

47 

74 

* 

1463 

83 

22 

4614 

2634 

32 

36 

2123 

7843 

1785 

21 

41 

4679 

2183 

6823 

183   250   6802   ]4640   18324 

*  The  process  is  twelve  six,  one  four  naught ;  the  40  is  put  down  and  the  1 
carried  to  the  units  column  in  the  next  pair,  then  ten  naught,  one  four  six. 

Any  person,  who  will  PBACTICE  this  method,  may 
add  two  columns  with  perfect  ease:  there  is  no  royal 
road  to  this  accomplishment :  speed  with  precision  can 
be  attained  only  by  persistent  PRACTICE. 

.Fives  are  always  easy  to  add ;  so  are  9's,  when  it 
is  borne  in  mind  that  adding  9  to  a  sum  places  it 
in  the  next  higher  ten  with  the  unit  1  less;  thus, 
17  +  9  =  26 :  89  -f  9  ^  48 ;  63  +  9  =  72. 


18  HOWARD'S  ART  OF  COMPUTATION. 


In  adding  long  columns  of  figures,  write  in 
e   margin,   lightly    with   pencil,  opposite   the 
st  figure  added,  the  unit  figure  of  the  sum 
immediately  exceeding  100.     By  doing  this  the 


g  mind  is  never  burdened  with  numbers  beyond 
7  100;  and  if  interrupted  in  the  work,  it  can  be 
4  resumed  at  the  stage  at  which  the  interruption 

6  occurred.     The  example  in  the  margin  shows 

7  the  method;  opposite  the  figure  7;  the  2  indi- 
9  eating  the  column,  so  far,  with  the   7  included, 

o 

9  amounts  to  102. 

INSTANTANEOUS  ADDITION  BY  COMBINATION. 

Write  two,  three,  four,  or  more  rows  of  miscel- 
laneous figures,  then  write  such  figures  as  will 
make  an  equul  number  of  nines  in  each  column; 
under  these  again,  write  another  row  of  miscella- 
neous figures. 

EXAMPLE  — 

4987 
47  3  G 
2187 

5012  one  9. 
5263  two  9's. 
7812  three  9's. 
4986 


RULE. — Bring  down  the  last  row,  less  the  num- 
ber of  nines  in  each  column,  and  prefix  the  number 
of  nines. 

*This  example  has  threo  nines  In  each  column. 


SUBTRACTION.  19 

RULE. — Write  the  numbers  so  that  the  units  in 
the  subtrahend  shall  be  directly  under  the  units  of 
the  same  value  in  the  minuend  ;  under,  and  in  the 
same  order,  write  the  difference. 

Subtract  473  from  1694.  1694 

473 

.i»a;i 

To  prove  Subtraction,  add  the  difference  to  the 
subtrahend;  if  correct,  their  sum  =  the  minuend. 

MULTIPLICATION. 

The  base  of  our  system  of  notation  is  10;  there- 
fore numbers  increase  and  diminish  in  a  tenfold 
ratio ;  increasing  from  the  decimal  point  to  the  left, 
and  decreasing  from  the  decimal  point  to  the  right ; 
hence  to  multiply  any  number  by  10,  annex  a 
cipher,  or  remove  the  point  one  place  to  the  right. 
To  multiply  any  number  by  100,  annex  two  ci- 
phers, or  remove  the  point  two  places  to  the 
right.  To  multiply  any  number  by  1000,  annex 
three  ciphers,  or  remove  the  point  three  places  to 
the  right. 

To  find  the  product  of  two  numbers,  when  the 
multiplicand  and  the  multiplier  each  contain  but 
two  figures. 

EXAMPLE  1 — 

3  3 
22 

726 


20  HOWARD' H  ART  OF  COMPUTATION. 

EXPLANATION — set  down  the  smaller  factor  under 
the  larger,  units  under  units,  tens  under  tens. 
Multiply  the  units  of  the  multiplicand  by  the  unit 
figure  of  the  multiplier;  thus,  2x3  =  6,  set  the 
6  down  in  unit's  place;  multiply  the  tens  in  the 
multiplicand  by  the  unit  figure  in  the  multiplier, 
and  the  units  in  the  multiplicand  by  the  tens  figure 
in  the  multiplier;  thus,  3x2  =  6,  and  3x2  =  6, 
add  these  two  products  together;  6  and  6  are  12; 
set  down  2,  carrying  the  ten  to  the  next  product, 
then  multiply  the  tens  in  the  multiplicand  by 
the  tens  in  the  multiplier;  thus,  3x2  =  6;  add 
the  one  carried  from  the  last  product,  making  the 
whole  product  726. 

The  same  method  can  be  applied  when  the  in  al- 
tiplicand  has  three  or  more  figures. 

EXAMPLE  2 — 

I  6  3 
2  4 


3912 

The  steps  are:  3  X  4  =12,  set  dow'n  the  2  and 
earry  the  1  ;  (6  x  4)  +  (3x2)  +  1  =  31;  set 
down  the  1,  and  carry  the  3.  (1  X  4)  +  (6  X  2)  -K  3 
=  19;  set  down  9  and  carry  1 ;  1  X  2  +  1  =  3, 
which  place  at  the  head  of  the  line,  making  a  total 
of  3912. 

When  the  multiplier  can  be  resolved  into  two 
factors,  it  is  sometimes  shorter  to  multiply  by  each 
factor,  than  by  the  whole  number. 

EXAMPLE,  multiply  163  by  24. 

8  X  3  =  24. 


MULTIPLICATION.  21 

1  6  3 


1304 
3 

3912.  Ans. 

When  the  multiplier  is  any  number  between  11 
and  20,  the  process  is  simply  to  multiply  by  the 
unit  of  the  multiplier,  set  down  the  product  under, 
and  one  place  to  the  right  o/,  and  then  add  to  the 
multiplicand. 

EXAMPLE,  multiply  1496  by  17. 

1496 
10472 


254  3  2.  Ans. 


or  thus :  1496 

1  7 


25432 

The  process  in  the  last  example  is : 

6x7=  42,  set  down  2  and  carry  4. 
9x7  +  6  +  4  =  73;  carry  7. 
4x7  +  9+7=  44;  carry  4. 
1X7  +  4  +  4  =  15;  carry  1. 
1+1=2. 

To  multiply  two  figures  by  11. 

RULE. — Between  the  two  figures  write  their  sum : 
thus:  multiply  43  by  11.     Ans.  473.     The  sum  of 


22          HOWARD'S  ART  OF  COMPUTATION. 

4  and  3  is  7 ;  place  the  seven  between  the  4  and  3, 
for  the  product. 

NOTK.— Add  one  to  the  hundreds  when  the  sum  exceeds  9. 

To  multiply  any  number  by  11. 

RULE— Bring  down  the  extreme  right  hand  figure, 
then  add  the  right  hand  figure  to  the  next,  and  bring 
down  the  sum ;  then  add  the  second  figure  to  the 
third  and  bring  down  the  sum,  adding  in  the  figure 
carried,  in  each  case,  and  so  on  to  the  end. 

EXAMPLE—             12345678 
11 

135802458 

To  multiply  any  two  numbers  ending  with  5. 

RULE. — Add  J  the  sum  of  the  figures  preceding 
the  5  in  each  number  to  the  product  of  the  same 
figures,  and  annex  25. 

NOTE.— When  the  sum  oi' the  preceding  figures  Is  an  odd  number,  add 
hall1  the  number  next  smaller  than  the  sum  and  annex  75. 

Multiply  85  by  65  and  105  by  35. 

85x65=7+8X6  with  25  annexed =5525 
105x35=6+10x3  "    75         "     =3675 

To  multiply  ivJien  t/ie  unit  figures  added,  equal  10, 
and  the  tens  are  alike,  as  67  X  63. 

RULE. — Multiply  the  units  and  set  down  the 
result,  then  add  one  to  the  upper  number  in  tens 
place,  and  multiply  by  the  lower. 


MULTIPLICATION.  23 

To  multiply  unlike  numbers  greater  than  a  com- 
mon base. 

RULE. — To  the  common  base  add  the  differences ; 
multiply  the  sum  by  the  base  and  add  the  product 
of  the  differences. 

EXAMPLE.— Multiply  603  by  612 

603+12  x  600+3x12-369,036. 
To  multiply  unlike  numbers  less  than  a  common 
base. 

RULE. — To  the  multiplicand  add  the  tens  and 
units  of  the  multiplier,  less  the  last  1  to  carry,  mul- 
tiply the  sum  by  the  common  base  and  add  the 
product  of  the  differences. 

EXAMPLE.— Multiply  93  by  89  and  293  by  289. 
89  293 

93  89 


8277  282x300+11x7=84,677. 

The  product  of  any  two  numbers— the  square  of 
their  mean,  diminished  by  the  square  of  half  their 
difference. 

EXAMPLE.— Multiply  22  by  18. 
202— 22  =  396. 

To  multiply  two  numbers  having  a  common  base, 
one  ending  ivith  25,  the  other  ending  with  75. 

RULE. — Multiply  the  common  base  by  one  more 
than  itself  and  annex  1875. 

EXAMPLE.— Multiply  675  by  625. 

6X7  with  1875  annexed=421,875. 
To  multiply  two  numbers  when  either  has  one  or 
more  ciphers  on  the  right,*as  26  by  20, 244  by  200,  etc. 


24  HOWARD'S  ART  OF  COMPUTATION: 

RULE. — Take  the  cipher  or  ciphers  from  one 
number  and  annex  it,  or  them,  to  the  other,  multi- 
ply by  the  number  expressed  by  the  remaining 
figures. 

EXAMPLE  1. — Multiply  20  by  20.     Ans.  520. 

Process. — 260  x  2  =  520. 
2.— Multiply  244  by  200.     Ans.  48800. 
24400  x  2  =  48800. 

RAPID  METHOD  OF  SQDARIM  NUMBERS, 

BY  THE   DIFFEREXCE  OF  A  NUMBER  AXD  ITS  BASJ;. 

For  squaring  a  number  greater,  than  Us  base. 

RULE. — To  the  given  number  add  the  differ- 
ence, multiply  the  sum  by  the  base ;  to  the  pro- 
duct add  the  square  of  the  difference. 

NOTE.    Take  the  nearest  convenient  multiple  of  ten  for  the  base. 

EXAMPLE  1. — What  is  the  square  of  11  ?  Ans.  121. 
Process. — Taking  10  for  the  base,  the  difference 
is  one  (1  +  lljx  10  +  I2  =  121. 

NOTE.  Until  this  rule  is  thoroughly  understood,  the  learner 
should  limit  his  exercises  to  numbers  near  10,  100,  1000,  Ac. ;  and 
then  operate  with  more  complex  numbers. 

1.       (22) 2  =  484. 

Process. — Taking  20  for  the  base,  the  difference 
(2+  22)  x  20  +  22  =  484. 

2._  (33)2  -  1089 


SQUARING  NUMBERS.  95 

For  squaring  numbers  less  'than  the  'base. 

RULE.  —  From  the  number  to  be  squared  subtract 
tie  difference,  multiply  the  result  by  the  base,  to 
the  product  add  the  square  of  the  difference, 

1.  (9)8  =  81. 

Process.  —  Taking  10  for  the  base,  the  difference 
or  complement  is  1.  then  (9—1)  X  10  +  I2  =  81. 

NOTB.  Iii  squaring  numbers  between  60  and  60,  take  50  for  the 
base;  to  25  add  the  difference,  call  the  stun  hundreds,  to  this  add 
the  square  of  the  difference. 


s  -  2601. 

Process.—  25  +  1  =  2600  +  I3  x  -  2601. 
2.—  (52)2  =  2704 

NOTB.  In  squaring  numbers  between  40  and  50;  to  15  add  the 
unit  figure,  call  the  number  hundreds,  to  the  sum  add  the  square 
of  the  difference,  taking  50  for  the  base, 

1.—  (41)*  =  1681. 

Process.—  15  +  1  =  1600  +  92  .=  1681. 

2.—  (42)  2  =  1764. 

3.__(43)s  —  1849. 

By  this  rule  the  squares  of  all  numbers  up  to  1000,  and 
larger  numbers  near  the  multiples  of  10  may  be  found 
with  less  labor  than  is  required  to  find  them  in  tables; 

The  square  of  any  number  ending  with  25=half 
the  number  of  hundreds  -j-  the  square  of  the  num- 
ber of  hundreds  X  10,000+625. 

3+62XlO,000+252=390,625:=625'* 


26  HOWARD'S  ART  OF  COMPUTATION. 

In  squaring  very  hi'gh  numbers,  use  the  foregoing 
rule  in  connection  with  the  following  formula: 

"The  square  of  any  number = the  sum  of  the 
squares  of  its  parts,  plus  twice  the  product  of  each 
part  by  the  sum  of  all  the  others." 

EXAMPLE.— Find  the  square  of  823,732 

823,0002^677,329,000,000 

823,000x732x2=     1,204,872,000 

7322= 535,824 

678,534,407,824 

WJien  either  the  tens  or  the  units  are  alike. 
RULE. — Multiply  the  units,  set  down  the  unit 
figure  of  the  product;  multiply  the  sum  of  the  un- 
like figures  by  one  of  the  like  figures,  then  multiply 
the  tens  figures  together,  adding  the  carrying  fig- 
ures as  you  proceed. 

Multiply  92  by  97  and  74  by  24. 

97  74 

_92  _24 

~8924  1776 

W7ien  the  units  are  alike  and  the  sum  of  the 
tens  is  ten. 

RULE. — Add  one  of  the  units  to  the  product  of 
the  tens,  and  annex  the  product  of  the  units. 

Multiply  74  by  34. 

7X3+4  with  16  annexed -2516. 

To  multiply  any  two  numbers  between  10  and  20. 

RULE. — To  the  product  of  the  units  prefix  1,  and 
add  the  sum  of  the  units  calling  it  tens. 

Multiply  18  by  14. 
8x4  with  1  picfixed  =  132.      132+12  tens,=252. 


DIVISION.  27 

When  the  multiplier  is  a  number  near,  and  less, 
than  a  multiple  of  10. 

RULE. — Annex  to  the  multiplicand  as  many  ci- 
phers as  there  are  in  the  next  order  of  tens  higher 
than  the  multiplier,  subtract  the  product  of  the  mul- 
tiplicand by  the  complement. 

Multiply  222  by  93. 

22,200—222x7=20,646. 

When  both  numbers  have  a  cipher  in  the  tens  place. 
RULE. — Write  the  product  of  the  units,  then  the 
sum  of  the  products  of  the  upper  hundreds  by  the 
lower  units,  and  the  lower  hundreds  by  the  upper 
units,  prefix  the  product  of  the  hundreds. 
Multiply  409  by  704. 

704 
409 

287936 

DIVISION. 

DIVISION  is  the  process  of  finding  how  many  times 
one  number  or  quantity  is  contained  in  another. 

RULE. — To  the  left  and  in  a  line  with  the  divi- 
dend, write  the  divisor,  separated  by  an  arc.  Take 
so  much  of  the  dividend  as  contains  a  number  less 
than  ten  times  the  divisor  ;  the  number  of  times  the 
divisor  is  contained  in  that  part  of  the  dividend  is 
the  first  figure  in  the  quotient;  annex  the  next 
unused  figure  of  the  dividend  to  the  remainder  to 
find  the  second  figure  of  the  quotient,  and  so  on  to 
the  end. 


28          HOWARD'S  ART  OF  COMPUTATION. 

Divide  49654809  by  4. 

4)49654809 


Ans.    12413702^ 

Process — The  divisor  4  is  contained  in  the  first 
figure  of  the  dividend  once,  therefore  1  is  the  first 
figure  in  the  quotient:  4  is  contained  twice  and  1  re- 
mainder in  9 ;  2  is  then  the  second  figure  in  the 
quotient :  the  next  unused  figure  6  annexed  to  the 
remainder  1=16:  4  is  contained  in  16  four  times, 
and  so  on  to  the  end. 

Divide  7983204  by  23. 

23)7983204(347095^ 
108" 
163 


220_ 

"134 

19 


Process.  79 — 23x3,  the  remainder  is  10;  the 
next  unused  figure  in  the  dividend  8,  annexed  to 
10=10&;  108— 23x4,  the  remainder  is  16;  to 
this  remainder  annex  the  next  unused  figure  in  the 
dividend,  and  so  on  until  the  quotient  is  complete. 
When  the  divisor  is  a  composite  number,  divide  by 
its  factors. 

EXAMPLE.— Divide  504  by  42.     42=7x6. 
6504 

7  JB£ 

12  Ans. 


FRACTIONS. 


FKACTIONS. 

GENERAL  PRINCIPLES  OF  FRACTIONS. 

% 

Multiplying  the  numerator,  multiplies  the  frac- 
tion. 

Dividing  the    numerator,  divides  the  fraction. 

Multiplying  the  denominator,  divides  the  frac- 
tion. 

Dividing  the  denominator,  multiplies  the  frac- 
tion. 

Multiplying  or  dividing  both  terms  of  the  frac- 
tion by  the  same  number,  does  not  ghange  its 
value. 

Fractions  are  'called  similar  when  they  have  a 
common  denominator,  as  |,  |- ,  f ,  |. 

Dissimilar  fractions  are  fractions  that  are  not 
alike,  as  |,  *,  f,  -J. 

The  numerators  of  similar  fractions  only  can  be 
added. 

The  common  denominator  is  written  under  the 
sum  or  difference. 

To  reduce  a  fraction  to  its  simplest  f  own. 

RULE. — Divide  both  terms  by  their  greatest  com- 
mon divisor  or  its  factors,  the  simplest  form,  or 
lowest  term  of  £  f,  is  obtained  by  dividing  both 
terms  by  12,  ff  =  f. 


30          HOWARD'S  ART  OF  COMPUTATION. 

To  find  the  greatest  common  divisor  of  two  numbers : 
RULE. — Divide  the  greater  by  the  less,  arid  the 
previous  divisor  by  the  remainder,  and  so  on  until 
there  is  no  remainder ;  the  last  divisor  is  the  answer. 
Find  the  greatest  common  divisor  of  18  and  27. 
18)27(1 
18 
0)18(2 

18  Ans.  9. 

To  find  the  least  common  multiple: 

RULE. — Cancel  all  the  numbers  that  are  con- 
tained in  any  of  the  others;  divide  all  those  not 
canceled  by  any  number,  or  the  greatest  of  its  fac- 
tors, that  will  exactly  divide  any  one  of  them,  bring 
down  eaeh  quotient  with  the  undivided  numbers 
and  proceed  as  before,  until  no  two  numbers  have 
a  common  divisor;  the  product  of  all  the  divisors 
and  the  remaining  numbers  is  the  answer. 

Find  the  least  common  multiple  of  36,  8,  9,  10, 
12,25,84,75.    Ans.  12x3x2x7x25=12600. 
12)36,  S,  9,  10,  19,  gg,  84,  75 
3,  2,        ?,  7,  25 

ADDITION   OF    FRACTIONS. 

RULE. — Make  the  fractions  similar  by  reducing 
them  to  the  same  denominator;  add  the  numera- 
tors, and  place  the  sum  over  the  common  denomi- 
nator. 

1.  What  is  the  sum  of  -|  and  £?  Ans.  fj, 

2.  What  is  the  sum  of  f  and  £?          Ans.  1TV 


FRACTIONS.  31 

SUBTRACTION    OF    FRACTIONS. 

RULE.  —  Make  the  fractions  similar  by  reducing 
them  to  the  same  denominator,  and  write  the  dif- 
ference of  the  numerators  over  the  common  de- 
nominator. 

1.  From  f  take  -^  Ans.  j> 

Process,  £  =  f  ,  f  —  f  =  £. 

2.  From  9J-  take  4£.  Ans.  4J-. 

3.  From  8£  take  3J.  Ans.  5^. 

4.  From  18|  take  3$.  Ans.  15T87. 

MULTIPLICATION    OF    FRACTIONS. 

RULE.  —  Multiply  the  numerators  together  for  a 
new  numerator,  and  the  denominators  together  for 
a  new  denominator. 

EXAMPLE.  —  Multiply  £  by  f. 


General  rule  for  multiplying  fractions  and  all 
mixed  numbers. 

RULE.  —  Multiply  the  whole  numbers  together, 
then  multiply  the  upper  whole  number  by  the 
lower  fraction,  then  multiply  the  uppei  fraction  by 
the  lower  whole  number,  then  multiply  the  frac- 
tions together,  and  add  all  the  products  together. 

1.  Multiply  8£  by  4£.  Ans.  36|. 


32  HOWARD'S  ART  OF  COMPUTATION. 

When  the  multiplier,  or  divisor  is  an  aliquot 
part  o/lOO  or  1000,  the  process  may  be  shortened 
by  the  use  of  the 

TABLE  OF  ALIQUOT  PARTS. 

12£  is  I  part  of  100.  8£  is  T^  part  of     100 

25  is  f  or  J  of  100.  16f  is  T*y  or  \  of    100 

37|  is  f  part  of  100.  33£  is  T%  or  \  of    100 

50  is  |  or  |  of  100.  66f  is  T^  or  f  of    100 

62|  is  |  part  of  100.  83$  is  {f  or  f  of    100 

75  is  |  or  f  of  100.  125    is  £  part  of  1000 

87£  is  £  part  of  100.  250    is  f  or  £  of  1000 

-6£  is  TV  part  of  100.  375    is  |  part  of  1000 

18f  isT3g-partof  100.  625    is  f  part  of  1000 

100.  875    is  \  part  of  1000 


To  multiply  by  the  aliquot  part  of  100. 

NOTE.  —  If  the  multiplicand  is  a  mixed  number,  reduce  the  fraction 
to  a  decimal. 

RULE.  —  Multiply  by  100,  by  annexing  two  ci- 
phers; such  part  of  the  product  as  the  multiplier 
is  part  of  100  will  be  the  answer. 

EXAMPLE.—  Multiply  86  by  12|.        Ans.  1075. 

To  divide  by  the  aliquot  part  of  100  or  1000. 

RULE.  —  Reduce  the  fraction,  if  any,  to  a  decimal, 
remove  the  point  two  places  to  the  left  for  100, 
three  places  for  1000  and  multiply  the  quotient  by 
the  part  the  divisor  is  of  100  or  1000. 

47825---100x8=47825-^12i=3826. 


FRACTIONS.  33 

To  multiply  any  two  numbers  together,  ending  with 
I,  as  9i  by  3£. 

RULE. — To  the  product  of  the  whole  numbers, 
add  half  their  sum,  plus  ^. 

NOTE.  When  the  sum  is  an  odd  number  take  half  the  next  num- 
ber below  it,  and  the  fraction  in  the  answer  will  be  %. 

1.  What  will  9J  Ibs.  of  rice  cost,  at  3^-  cts.  per 
Ib?     Ans.  33 J  cents. 

Process. — The  sum  of  9  and  3  is  12;  half  this 
sum  is  6;  then  we  say  9  times  3  is  27,  and  6  =  33; 
to  this  add  £. 

2.  What  will  9J  doz.  buttons  cost,  at  8£  cts.  per 
doz  ?     Ans.  80f  cts. 

3.  What  will  11|  Ibs.  of  beef  cost,  at  9£  cents 
$er  Ib?     Ans.  $1.09^. 

4.  What  will  7-J-  doz.  eggs  cost,  at  13  J  cents  per 
doz?     Ans.  $1.01  J. 

To  multiply  any  two  numbers  together  having  the 
same  fraction. 

RULE. — To  the  product  of  the  whole  numbers, 
add  the  product  of  their  sum  by  the  fraction;  to 
this  add  the  product  of  the  fractions. 

1.  What  will  13f  Ibs.  of  beef  cost,  at  7J  cents 
per  Ib?  Ans.  $1.06-^. 

Process. — The  sum  of  13  and  7  is  20,  three-fourths 
of  this  sum  is  15,  so  we  say,  7  times  13  is  91,  and 
15  =  106.  to  which  add  the  product  of  the  fractions, 
(T\)  and  the  result  js  the  Ans.  $1.06TV 


34          HOWARD'S  ART  OF  COMPUTATION. 

In  actual  business  calculations,  any  fraction  less 
than  a  cent  is  reckoned  as  one  cent;  therefore  in 
dealing  with  such  questions  as  13J  pounds  of  beef 
at  7J  cents  a  pound,  it  is  sufficiently  accurate  to  say : 

16oM3=3.     £  of  7=2.     13><7+3+2 =96  cents; 

Or  17  J  Ibs.  of  cheese  at  9  J  cents  per  pound. 
J  of  17=6.     i  of  9=2.     ffx9+6+2=$1.61. 

When  the  whole  numbers  are  alike,  and  the  sum  of 
the  fractions  is  a  unit. 

RULE. — Take  the  product  of  the  whole  numbers, 
to  this  add  the  integer  in  the  multiplicand,  then  add 
the  product  of  the  fractions,  and  the  result  will  be 
the  answer. 

1.  Multiply  2|  by  2£.  Ans.  6J. 

* 
Process— 2  x2  +  2  =  6+|Xi  =  6£. 

2.  31  X  by  3f  =  12f. 

3.  7|  X  ?i  =  56^- 

4.  9f  X  9f  =  90|f. 

5.  19|  X  19f  =  380tf 

6.  101 1  X  101-i  =  10302^. 

7.  109TV  X  109T43  =  11990T\V 

8.  98TV  X  98^  =  9702TW 

9.  96}  X  96f  =  9312|f 

10.  9947||  X  9947T6T  =  98952756^. 

11.  9995714  X  99957JV  =  9,991,501,806TV^. 


DECIMALS.  35 

DIVISION  OF    FRACTIONS. 

RULE. — Reduce  whole  and  mixed  numbers  to  the 
form  of  an  improper  fraction.  Multiply  the  divi- 
dend by  the  divisor  inverted. 

Divide  8  by  1J.  Ans.  65. 

Process — 1^  inverted  is  5.     5X1=^=65. 

To  divide  by  any  number  expressed  by  1  and  any 
number  of  ciphers,  remove  the  decimal  point  as  many 
places  to  the  left  as  there  are  ciphers  in  the  divisor. 

74864-1000=74.864 

DECIMALS. 

The  system  of  Decimal  fractions  is  so  pre-eminently 
simple,  that  when  it  is  generally  understood  it  will 
entirely  displace  the  clumsy  system  of  common  frac- 
tions. In  harmony  with  our  system  of  notation,  it  is 
a  fraction  always  having  some  power  of  ten  for  a 
denominator:  thus  .1  =  TV,  .03  =  T3<j,  -007  —  Woo, 
47.8  =  47  T%,  &c.,  &c. 

Where  common  fractions  occur  the  calculation  may 
be  often  simplified  by  reducing  them  to  decimals.  To 
reduce  a  common  fraction  to  a  decimal. 

RULE. — Divide  the  numerator  ~by  the  denominator. 

|  =  .5       |  =  .25      £  =  .1 25    TV  =  .0625. 
|  =  .75     |  =  .3333    §  =  .66M     i  s=  .2  *  =  .4 

=  .8       »  =  .6  '         =  .1666        =  .II11    TV  =  -08333 


36  HOWARD'S  ART  OF  COMPUTATION. 

ADDITION  AND  SUBTRACTION  OF  DECIMALS 
Are  performed  in  the  same  manner  as  in  ichole  numbers  ; 
care  being  taken  to  properly  point  off  I  he  decimal  places, 

MULTIPLICATION  OF  DECIMALS. 

Rule. — Multiply  as  in  whole  numbers,  and  point  off 
as  many  places  to  the  left  for  decimals  as  there  are  dec- 
imal places  in  both  factors. 

1.  Multiply  .5  by  .5.  Ans.  .25. 

2.  Multiply  1.75  by  .3.  Ans.  .525 

3.  27.46  by  .4  Ans.  10.984 
To  multiply  by..  1  remove  the  decimal  point  one 

place  to  the  left,  by  .01  ft#o  places,  by.OOl  three  places, 
b}'  10  one  place  to  the  right,by  100  two  places,  by 
1000  three  places,  &c  ,  &c. 

Note. — In  practical  business  the  answer  to  three  decimal  pla- 
ces is  sufficiently  exact,  the  third  decimal  only  counting  for  mills, 
the  drudgery  of' finding,  and  writing  the  figures  for  decimals  of 
no  value,  may  be  avoided  by  reversing  the  order  of  the  multiplier 
and  writing  the  first  figure  of  the  reversed  multiplier  under  the 
third  decimal  figure  in  the  multiplicand,  begin  each  line  of  the 
partial  products,  with  the  product  of  the  multiplying  figure 
and  the  figure  directly  above  it,  adding  the  carrying  figure,  if 
any,  from  the  immediate  right  hand  figure. 

What  is  the  par  value  in  American  gold  coin  of 
£11  „  4  „  3,  Sterling? 

£11.2125  11.2125 

4.8665  56  684 

"~560"625~  44850 

672750  8  970 

672750  673 

897000  67 

448500  5 


$54.56563125  $54.565 

This  example  illustrates  the  difference  of  the  two  methods. 


DECIMALS.  37 

When  there  are  .not  as  many  figures  in  the  pro- 
duct as  there  are  decimals  in  both  factors,  supply 
the  deficiency  by  prefixing  ciphers. 

1.  Multiply  .3  by  .3.  Ans.  .09. 

2.  Multiply  .29  by  .004.  Ans.  .00116. 

DIVISION    OF    DECIMALS. 

The  division  of  decimals  is  performed  in  the  same 
manner  as  in  whole  numbers,  care  being  taken  to 
point  off  the  decimal  places  in  the  quotient. 

RULE. — Divide  as  in  whole  numbers,  and  point 
off  in  the  quotient  as  many  places  to  the  left  for 
decimals  as  the  decimal  places  in  the  dividend 
exceed  those  in  the  divisor. 

Divide  .244  by  .4.  Ans.  .61. 

Divide  .255  by  .05..  Ans.  5.1. 

The  learner  can  supply  additional  examples  at 
discretion,  bearing  in  mind  the  following:  The 
dividend  must  always  contain,  at  least,  as  many 
decimal  places  as  the  divisor.  When  the  number 
of  figures  in  the  quotient  is  less  than  the  excess  of 
the  decimal  places  in  the  dividend  over  those  in  the 
divisor,  the  deficiency  must  be  supplied  by  prefix- 
ing ciphers.  When  there  is  a  remainder  after 
dividing  the  dividend,  annex  ciphers,  and  continue 
the  division;  the  ciphers  annex?d  ^re  decimals  to 
the  dividend. 


38 


HOWARD  S    ART    OF    COMPUTATION. 


•When  tlie  divisor  is  a  quantity  a  little  less  than  a  number 
expressed  by  1  and  one  or  more  ciphers: 

RULE.  —  Divide  by  the  nearest  higher  number  ex- 
pressed by  unity  and  one  or  more  ciphers  ;  multiply  the 
quotient  by  the  difference  of  the  assumed  and  the  given 
divisor,  writing  the  product  under,  and  as  many  places 
to  the  right,  as  there  are  significant  figures  in  the  given 
divisor;  repeat  this  operation  with  each  succeeding  quo- 
tient as  often,  and  to  as  many  decimal  places  as  the  an- 
swer requires;  the  sum  of  the  quotients  is  the  answer. 

1264.86568 

EXAMPLE—  Divide  2'l<>648 

126486568  by  99000.  -12g 


_ 

1277.64208  Ans. 

The  answer  to  this  exarnple  may  be  found  by  removing 
the  point  three  places  to  the  left  and  dividing  by  9x11. 


126486.568 


14054.0631 


1277.6421 


The  labor  of  finding  the  answer  to  valueless  decimals 
may  be  saved  by  cutting  off  a  figure  from  the  right  hand 
of  the  divisor,  as  each  new  figure  in  the  quotient  is  found, 
carrying  what  would  have  been  obtained  by  the  multi- 
plication of  the  figure  cut  off,  1  if  the  multiplication  pro- 
duces more  than  5  and  less  than  15,  2  if  more  than  15 
and  less  than  25,  etc. 


73.412)648.7654386(8.8373 
587.296 


58729 

4 
6 
83~ 
36 

2739 
2202 

537 
513 
23 
22 

478 
884 
5946 
0236 

1 

5740 

73.412)648.7654386(8.8373 

587.296 

"61469 

58730 

2739 

2202 

537 
514 

•       .23 

22 

1 


PROPORTION.  39 

PROPORTION. 

Proportion  is  the  equality  of  ratios. 

Ratio  is  the  relation  which  one  quantity  bears  to 
another  of  the  same  kind,  with  reference  to  the 
number  of  times  that  the  one  is  contained  in  the 
other. 

Thus,  the  ratio  of  7  to  21  is  3,  because  7  is  con- 
tained 3  times  in  21,  or  21  is  3  times  seven.  The 
same  result  is  obtained  if  we  divide  7  by  21, .for 
we  then  find  ¥7T  =  £,  which  means  that  7  is  £  of  21, 
and  this  expresses  the  very  same  relation  as  before, 
to  say  that  7  is  -J-  of  21  is  precisely  the  same  as  to 
say  that  21  is  3  times  7.  The  ratio  of  9  to  27  is 
3,  but  we  have  seen  that  the  ratio  of  7  to  21  is  also 
3,  therefore,  the  ratios  of  7  to  21  and  9  to  27  are 
the  same,  21  -~-  7  =  27  -f-  9,  and  these  quantities 
are  thefore  called  proportionals. 

In  any  proportion,  as 

7:  21::  9:  27 

the  product  of  the  middle  numbers,  21  and  9, 
equals  the  product  of  the  extremes,  7  and  27 : 
hence  the  rule,  that  when  the  fourth  proportional 
is  unknown, 

Multiply  the  second  and  third  terms,  and  divide 
the  product  by  the  first. 

EXAMPLE. — If  7  sheep  cost  21  dollars,  what  will 
9  cost  at  the  same  rate  ?  27  dollars,  Ana. 


40  HOWARD'S  ART  OF  COMPUTATION. 

2d  term,        21  3 

3d  term,          9         Or  thus,  #L  X  9  =  27 


1st  term,  7)189  f 

27 

Proportion  is  so  much  used  in  business,  and  may 
be  simplified  and  shortened  so  much  by  the  fore- 
going process  of  cancellation,  that  the  pupil  must 
learn  both  before  he  can  hope  to  be  expert  with 
business  calculations. 

CANCEL  ING  IN  CALCULATION. — Whenever  it  is  re- 
quired to  multiply  two  or  more  numbers  together, 
and  divide  by  a  third,  the  first  step  is  to  state  the 
problem  in  its  most  manageable  form ;  this  can 
only  be  done  by  the  use  of  the  arithmetical  signs. 

The  statement         28  X  1 2 


14 

is  to  be  read,  28  multiplied  by  12  is  to  be  divided 
by  14. 

Stating  the  problem  as  above  we  see  at  a  glance 
if  the  divisor  is  contained,  and  how  many  times,  in 
either  of  the  multipliers. 

In  the  foregoing  example  the  divisor,  14,  is  con- 
tained twice  in  the  multiplier,  28 ;  then  cancel  the 
14  and  substitute  2  for  the  28,  and  say,  twice  12  is 
24  the  answer. 

Process,  2 

n  x  12 

—  24. 

14 


PROPORTION.  41 

EXAMPLE. — It  9  turkeys  cost  $18,  what  will  be 
the  cost  of  27? 

3 

is  x  n 

=  $54,  Answer. 

$ 

If  the  divisor  is  not  contained  evenly  in  either 
of  the  multipliers,  there  may  be  a  common  divisor 
for  the  divisor  itself  and  one  of  the  multipliers ;  if 
so,  the  common  divisor  may  be  used  in  cancel  ing,, 
thus: 

7 

0?  X  8 
=  18*,  Ans. 


A  glance  shows  that  -9  is  the  common  divisor  for 
63  and  27. 

When  a  common  divisor  has  been  used  to  change 
the  expression  of  the  divisor  and  one  of  the  mul- 
tipliers, the  new  divisor  may  be  cancel  ed  when  it 
is  contained  an  even  number  of  times  in  the  other 
multiplier. 

EXAMPLE  —  7       2 

0?  X  $ 


Process  —  36  and  63  divided  by  9,  the  common 
divisor,  becomes  4  and  7  respectively,  the  4  into  8,  2 
times,  cancel  4  and  8,  and  twice  7  is  14,  the  answer^ 


42  HOWARD'S  ART  OF  COMPUTATION. 

Summary  of  the  rapid  process  for  cancel  ing. 

1.  Draw  a  horizontal  line ;  above  the  line  write 
dividends  only ;  below  the  line  write  divisors  only. 

2.  If  there  are  ciphers  above  and  below  the  line, 
erase  an  equal  number  on  either  side;  1  standing 
alone  may  be  disregarded. 

3.  If  the  same  number  stands  above  and  below 
the  line,  erase  them  both. 

4.  If  any  number  on  either  side  of  the  line  will 
divide  any  number  on  the   other  side  of  the  line 
without  a  remainder,  divide,  and  erase  the  two 
numbers,  retaining  the  quotient  figure  on  the  side 
of  the  larger  number. 

5.  If  any  two  numbers  on  either  side  have  a 
common  divisor,  divide  them  by  that  number,  and 
retain  the  quotients  only. 

6.  Multiply  all  the  numbers  above  the  line  for  a 
dividend,  and  those  below  the  line  for  a  divisor  j 
divide,  and  the  quotient  is  the  answer. 

7.  Write  all  the  terms  of  the  same  kind  in  units, 
or  fractions,  of  the  same  denomination*  i.  e.,  feet,  or 
fractions  of  a  foot ;    yards,  or  fractions  of  a  yard. 

EXAMPLE.— If  7  inches  of  velvet  cloth  cost  2£ 
dollar*,  what  will  be  the  cost  of  7  yards?  $90,  Ans. 

18 
5      K      H 

Process,  -  X  -  X  —  =  90. 

%      1       H 

XOTE.— 2£  dollars  =  f ,  7  yards  =  |,  7  inches  = 
£r  of  a  yard,  ^\  inverted  is  \6-. 


CANCELLATION.  43 

If  an  upright  line  is  used  put  dividends  on  the  right, 
and  divisors  on  the  left.  In  stating  a  question  put  the 
term  of  the  same  kind  as  the  required  term  first,  at 
the  top,  on  the  right  of  the  line ;  then  the  other  terms 
in  pairs  of  the  same  kind  ;  if  the  effect  is  to  increase- 
the  answer,  put  the  larger  term  on  the  right,  and 
vice  versa. 

EXAMPLE: — If  5  compositors,  in  16  days  of  14  hours 
long,  can  compose  20  sheets  of  24  pages  in  each  sheet,, 
50  lines  in  a  page,  and  40  letters  in  a  line,  in  how 
many  days  of  7  hours  long  may  10  compositors  com- 
pose a  volume  containing  40  sheets,  16  pages  in  a. 
sheet,  60  lines  in  a  page,  and  50  letters  in  a  line,  1  of 
the  second  set  of  compositors  being  equal  tt>  2  of  the 
first  ?  Ans.  16  days. 


Days    

Compositors  .  10 

Hours  7 

Sheets 20 

Pages  24 

Lines    50 

Letters    40 

Ratio    2 


16  required  term. 

5  less  time  with  10  than  5  men. 

14  more  days  with  7  than  14  hours  a  day.. 

40  more  time  to  set  40  than  20  sheets. 

6  less  time  to  set  16  than  24  pages. 
fiO  more  time  to  set  60  than  50  lines. 
50  more  time  to  set  50  than  40  letters. 

1 


NOTE.— Excepting  the  upper  term  16,  the  numbers  on  one  side  ex- 
actly balance  the  numbers  on  the  other,  and  may  all  be  canceled. 

This  method  acts  like  a  pair  of  scales,  we  use  known; 
to  find  the  value  of  unknown  quantities ;  the  arrange- 
ment of  the  terms  is  so  very  plain  and  natural  as  to 
be  easily  apprehended ;  by  its  use  the  most  complex 
problems  are  simplified,  and  all  business  calculations 
made  with  very  few  figures,  and  very  little  mental? 
effort ;  it  is  accurate,  and  free  from  the  risk  of  error. 


44          HOWARD'S  ART  OF  COMPUTATION. 

To  compute  Interest  by  Cancellation. 

1st,  on  the  right  of  the  line  write  the  principal, 
the  time  in  days,  and  the  rate  per  cent. 

2nd,  on  the  left  the  number  of  days,  or  its  factors, 
in  the  year,  and  remove  the  decimal  point  two  places 
to  the  left. 

Find  the  interest  on  £428.10  for  146  days  at  5 
per  cent,  per  annum  -of  365  days. 

tt  42.85    ) 

140    2  >  =8.57     =     <£8,,11,,5. 

99          ) 

Find  the  interest  on  $99  for  23  days  at  4  per 
cent,  per  annum  of  360  days. 


10 


4  V  =.253  =  25  cents,  3  mills. 

23 


In  computing  interest  at  rates  per  cent,  per  month, 
write  principal,  time  and  rate  as  above ;  write  3  on 
the  left  of  the  line  and  remove  the  decimal  point 
three  places  to  the  left. 

Find  the  interest  on  $348  for  24  days  at  1J  per 
cent,  per  month. 
348) 
24    V  =3.480     =     $3.48. 

5     I 

Legal  Interest  is  reckoned  on  the   basis    of  3G5 

days  to  the  year,  when  this  is  required,  and  the  cal- 
culation is  made  on  the  basis  of  360  days,  subtract 
7*3  for  the  common  year,  or  %\  for  a  leap  year,  and 
the  legal  interest  will  be  shewn;  about  1J  cents  for  each 
dollar  of  interest. 


RULES   FOB   FARMERS.  45 

RAPID 
RULES  FOR  FARMERS. 

The  practice  of  buying  or  selling  grain  by  the 
100  pounds,  or  the  cental  system,  is  becoming 
almost  universal,  and  has  many  advantages  over 
the  bushel. 

The  following  rules  for  finding  the  relative  values 
of  the  bushel  and  the  cental  are  easy  to  learn,  and 
true  and  rapid  in  execution. 

To  find  the  value  per  cental  tvhen  the  price  per 
bushel  is  given,. 

RULE. — Set  down  the  price  per  bushel;  remove 
the  decimal  point  two  places  to  the  right,  and 
divide  by  the  number  of  pounds  in  the  bushel. 

EXAMPLE. — If  wheat  is  $1.80  per  bushel,  what  is 
its  value  per  cental  ?  Ans.  $3. 

Process—  60)180 


To  find  the  value  per  bushel  when  the  price  per 
cental  is  given. 

RULE. — Set  down  the  price  per  cental ;  multiply 
by  the  number  of  pounds  in  the  bushel,  and  remove 
the  decimal  point  two  places  to  the  left. 


46  HOWARD'S  ART  OF  COMPUTATION. 

EXAMPLE. — If  wheat  is  $3.00  per  cental,  what  is 
the  value  of  a  bushel?  .  Ans.  1.80. 

Process —  3.0  0  0 

6 


1 .8  0  0  0 

RAPID    RULE    FOR    RECKONING    THE    COST    OF    HAY. 

RULE — Multiply  the  number  of  pounds  by  half 
the  price  per  ton,  and  remove  the  decimal  point 
three  places  to  the  left. 

EXAMPLE. — What  is  the  cost  of  764  Ibs.  of  hay  at 
$14  per  ton?  Ans.  $5.348. 

Process —  764 

=  7 


5.3  48 

NOTE.  '—The  above  rule  applies  to  anything  of  which  2,000  pounda 
is  a  ton. 

To  Measure  Grain. 

RULE. — Level  the  grain;  ascertain  the  space  it 
occupies  in  cubic  feet;  multiply  the  number  of 
cubic  feet  by  8,  and  point  off  one  place  to  the  left. 

EXAMPLE. — A  box  level  full  of  grain  is  20  feet 
long,  10  feet  wide,  and  5  feet  deep.  How  many 
bushels  does  the  box  contain  ?  Ans.  800  bush. 


RULES  FOR  FARMERS.  47 

Process— 20  X  10  X  5x8  —  10=  800. 
Or,  1  0  0  Oft. 

8 


8  00.0 

NOTE.— Exactness  requires  the  addition  to  every  one  hundred 
bnshela  of  .44  of  a  bushel 

The  foregoing  rule  may  be  used  for  finding  the 
number  of  gallons,  by  multiplying  the  number  of 
bushels  by  8. 

If  the  corn  in  the  box  is  in  the  ear,  divide  the 
answer  by  2,  to  find  the  number  of  bushels  of 
shelled  corn,  because  it  requires  two  bushels  of 
ear  corn  to  make  one  of  shelled  corn. 

RAPID  RULES  FOR  MEASURING  LAND 
WITHOUT  INSTRUMENTS. 

In  measuring  land,  the  first  thing  to  ascertain  is 
the  contents  of  any  given  plot  in  square  yards; 
then,  given,  the  number  of  yards,  find  out  the 
number  of  rods  and  acres. 

The  most  ancient  and  simple  measure  of  dis- 
tance is  a  step.  Now,  an  ordinary-sized  man  can 
train  himself  to  cover  1  yard  at  a  stride,  on  the 
average,  with  sufficient  accuracy  for  ordinary  pur- 
poses. 

To  make  use  of  this  means  of  measuring  dis- 
tances, it  is  essential  to  walk  in  a  straight  line ;  to 
do  this,  fix  the  eye  on  two  objects  in  a  line  straight 
ahead,  one  comparatively  near,  the  other  remote; 


48          HOWARD'S  ART  OF  COMPUTATION. 

and,  in  walking,  keep  these  objects  constantly  in 
line. 

Farmers  and  others  by  adopting  the  following  sim- 
ple and  ingenious  contrivance,  may  always  carry  with 
them  the  scale  to  construct  a  correct  yard  measure. 

Take  a  foot  rule,  and  commencing  at  the  base  of 
the  little  finger  of  the  left  hand,  mark  the  quarters 
of  the  foot  on  the  outer  borders  of  the  left  arm, 
pricking  in  the  marks  with  indelible  ink. 

To  find  the  area  of  a  Jour-sided  figure,  two  of 
ivhicJi  sides  are  parallel. 

RULE. — Multiply  the  length  and  the  breadth  to- 
gether, and  the  product  is  the  area. 

To  find  the  area  of  a  square,  square  one  of  its  sides. 

RULE. — When  the  length  of  two  opposite  sides 
is  unequal,  add  them  together,  and  take  half  the 
sum  and  multiply  by  the  breadth. 

EXAMPLE  1.  How  many  square  yards  in  a  square 
piece  of  land,  101  yds.  on  each  side? 

Process— 101 2  =  Ans.  10,201  yards. 

EXAMPLE  2.  How  many  yards  in  a  piece  of  land 
60  yards  long  and  20  yards  wide?  Ans.  1200. 

Process— 600  x  2  =  1200. 


RULES  FOR  FARMERS.  49 

EXAMPLE  3.  How  may  yards  in  a  piece  of  land, 
one  side  is  40  yards  long,  and  the  other  side  60 
yards  long,  parallel  sides  being  10  yards  apart? 

40  +  60  X  10 

Process,  =  500. 

2 

500  yards,  Ans. 

To  find  the  area  of  any  three-sided  figure. 

RULE. — Multiply  the  longest  side  into  one-half 
the  distance  from  this  side  to  the  opposite  angle. 

EXAMPLE. — What  is  the  area  of  a  triangular  plot 
of  land,  the  longest  side  of  which  is  80  yards,  and 
the  shortest  distance  from  this  side  to  the  opposite 
angle  40  yards  ? 

40x80 
Process,  —  1600  yds.  Ans. 

2 

To  find  how  many  rods  in  length  will  make  an  acre, 
the  width  being  given. 

RULE. — Divide  160  by  the  width,  and  the  quo- 
tient will  be  the  answer. 

EXAMPLE. — If  a  piece  of  land  be  4  rods  wide, 
how  many  rods  in  length  will  make  an  acre  ? 

160  -7-  4  =  40  rods  Ans. 


50  HOWARD'S  ART  OF  COMPUTATION. 

To  find  the  number  of  acres  in  any  plot  of  land, 
the  number  of  rods  being  given. 

RULE.  —  Divide  the  number  of  rods  by  8,  and  the 
quotient  by  2,  and  remove  the  decimal  point  one 
place  to  the  left. 

EXAMPLE.  —  In  6840  rods,  how  many  acres  ? 

42  f  acres  Ans. 
Process.—  8)6840 


~42?T5 

To  find  the  number  of  acres,  the  number  of  yards 
being  given. 

Divide  the  number  of  yards  by  4840  or  its  factors. 
EXAMPLE.  —  Find  how  many  acres  in  21,780  yds. 
21,780 


A  circle  encloses  the  largest  area  within  the  short- 
est fence. 

The  length  of  a  circular  fence  =  the  square  root 
of  the  area  X  1JX3}. 

Find  the  length  in  yards  of  a  circular  fence  to 
enclose  10  acres. 

^48-100=220.     220xlfcX3J=780  yards. 

A  square  plot  of  the  same  area  requires  a  fence  880 
yards  long. 

The  largest  area  enclosed  within  the  shortest  fence, 
hi  u  rectangular  plot,  ia  u  square. 

£93  J  yard*  offence  will  enclose  a  square  plot  of  two 
acres;  it  would  require  2  mile;  and  2  rods*  offence  to  en- 
same  area  in  a  rectangular  plot  1  rod  wide. 


RULES  FOR  MECHANICS. 


51 


RAPID 


RULES  FOE  MECHANICS. 

To  LAY  OFF  A  SQUARE  CORNER. — Measure  off 
eight  feet  from  the  end  of  one  sill,  and  there  make 
a.  mark ;  then  measure  off  six  feet  on  the  sill  lying 
at  right  angles  with  the  first,  and  make  another 
mark;  then  lay  on  a  ten  foot  pole,  one  end  of  it 
squarely  with  the  first  mark.  Move  the  sill  in  or 
out  until  it  exactly  squares  with  it.  The  figure  thus 
made  in  marking  off  the  sills,  and  in  the  laying 
down  the  ten  foot  pole  is  a  right  angle  triangle. 

8  ft. 


Another  method  for  laying  off  a  square  corner. 

Take  a  measure  and  lay  off  with  it  a  triangle, 
one  side  of  which  is  lour  feet  long,  another  three 
feet,  and  the  remaining  side  five  feet.  This  trian- 
gle will  be  right  angled,  and  the  two  shorter  sides 
will  serve  to  lay  off  an  exact  square. 


52          HOWARD'S  ART  OF  COMPUTATION. 

To  Measure  Grindstones,  or  any  Cylinder. 

KTTLE. — Multiply  the  square  of  the  radius  by  the 
thickness,  both  in  feet,  or  fractions  of  a  foot,  and  the 
product  by  3| ; 

or 

Multiply  the  square  of  the  diameter  by  the  thick- 
ness, both  in  inches,  and  divide  by  2200,  the  answer  is 
in  cubic  feet. 

EXAMPLE. — How  many  feet  m  a  grindstone  24 
inches  in  diameter  and  4  inches  thick  ? 

1st  Method.  2nd  Method. 

1  X  1  X  22  24  X  24  X  4 

=  1-04 =  1-04  ft. 

3x7  2x11 

Measure  of  Superfices  and  Solids. 

Superficial  measure  is  that  which  relates  to  length 
and  breadth  only,  not  regarding  thickness.  It  is 
made  up  of  squares,  either  greater  or  less,  according 
to  the  different  measures  by  which  the  dimensions 
of  the  figure  are  taken  or  measured.  Land  is 
measured  in  this  way,  its  dimensions  being  taken 
in  inches,  feet  and  yards,  or  links,  rods  and  acres. 
The  contents  of  boards  also,  are  found  in  this  way, 
their  dimensions  being  taken  in  feet  and  inches. 
The  standard  of  measure  is  as  follows:  12  inches 
in  length  make  one  foot  of  long  measure ;  therefore, 
12  x  12  =  144,  the  square  inches  in  a  superficial 
foot. 


RULES  FOR  MECHANICS.  53 

1.  If  the  floor  of  a  room  be  20  feet  long  by  18 
feet  wide,  how  many  square  feet  are  contained  in 
it?  Ans.  360  feet. 

Process— 180  X  2  —  360. 

2.  If  a  board  be  4  inches  wide,  how  much   in 
length  will  make  a  foot  square  ?     Ans.  36  inches. 

Process — 144  divided  by  the  width,  thus,  ij±  = 
36. 

3.  If  a  board  be  21  feet  long  and  18  inches  broad, 
how  many  square  feet  are  contained  in  it  ? 

Ans.  31£  sq.  ft. 

Process  —  Multiply  the  length  in  feet  by  the 
breadth  in  inches,  and  divide  the  product  by  12. 

3 

21  X  « 

=  31*. 

n 
2 

Or  thus,  18  inches  equals  1£  ft.;  21  x  H  =  31£. 

To  measure  a  board  wider  at  one  end  than  the 
other,  of  a  true  taper. 

RULE. — Add  the  widths  of  both  ends  together; 
halve  the  sum  for  the  mean  width,  and  multiply 
the  mean  width  by  the  length. 


54          HOWARD'S  ART  OF  COMPUTATION. 

EXAMPLE. — How  many  square  feet  in  a  board  20 
feet  long,  9  inches  in  width  at  one  end,  and  11 
inches  at  the  other?  Ans.  16f  sq.  ft. 

Process — 

9  +  11  20  X  10 

=  10  in.,  mean  width; —  16|k 

2  12 

To  find  the  board  measure  of  planks  and  joists. 

RULE. — Find  the  contents  of  one  side  of  the 
plank  or  joist  by  the  preceding  rule,  and  multiply 
the  result  by  the  thickness  in  inches. 

EXAMPLE.— What  is  the  board  measure  of  a  plank 
18  feet  long,  10  inches  wide,  and  4  inches  thick? 

Ans.  60  ft. 

18  X  10  X  4 

Process — =  60. 

12 

TJie  diameter  being  given,  to  find  tJie  circumference. 
RULE. — Multiply  the  diameter  by  3|. 

EXAMPLE. — What  is  the  circumference  of  a  wheel 
the  diam'eter  of  which  is  42  inches  ?  Ans.  1 1  ft. 


RULES  FOR  MECHANICS.  .% 

To  find  the  diameter  when  the  circumference  is 
given, 

RULE. — Divide  the  circumference  by  3^. 

EXAMPLE. — What  is  the  diameter  of  a  wheel,  the 
circumference  of  which  is  11  feet?  Ans.  3£  feet 

II        1 

Process —  —  X  —  =  3J 

1         n 
2 

What  is  the  width  of  a  circular  pond,  154  rods 
in  circumference?  49  rods  Ans. 

1?4         7 

Process —  X  —  —  49. 

1          £2 

The  diameter  being  given,  to  find  the  area. 
RULE. — Multiply  the  square  of  the  radius  by  87. 
Find  the  area  of  a  circle  36  inches  in  diameter. 

'  =7.07  feet. 

2X2X  7 

The  length  of  a  cylinder  is  equal  to  the  capacity 
—-the  square  of  the  radius-^-3  7. 

Find  the  depth  of  a  circular  cistern,  7  feet  wide, 
containing  2400  U.  S.  gallons. 

2400X2X2X2X  7 


15X7X7X22 


=8.31  feet. 


56          HOWARD'S  ART  OF  COMPUTATION. 

To  find  how  many  solid  feet  a  round  stick  of  tim- 
ber of  the  same  thickness  throughout,  will  contain 
when  squared. 

RULE.  —  Square  half  the  diameter,  in  feet,  mul- 
tiply by  the  depth,  and  then  by  2. 

Find  how  many  solid  feet,  when  squared,  in  a 
round  log  2J  feet  wide  and  10  feet  long. 


General  rule  for  measuring  timber  to  find  the 
solid  CQntents  in  feet. 

RULE.  —  Multiply  the  depth,  in  feet,  or  fractions 
of  a  foot,  by  the  breadth,  multiplied  by  the  length. 

How  many  solid  feet  in  a  piece  of  timber  2  feet 
wide,  10  inches  thick  and  12  feet  long. 
2x5x12 


6 


.=20  feet. 


To  find  the  contents  of  a  true  tapered  pyramid, 
whether  round,  square,  or  triangular. 

RULE.  —  Multiply  the  area  of  the  base  by  J  the 
height. 

How  many  cubic  feet  in  a  round  stick  of  timber, 
truly  tapering  to  a  point,  1|  feet  in  diameter  at  the 
base  and  24  feet  long. 


How  many  cubic  feet  in  a  square  block  of 


RULES  FOR  MECHANICS.  57 

marble,  truly  tapering  to  a  point,  24  inches  on  each 
side  at  the  base,  and  twelve  feet  high. 

24X24X4  ^  2X2X4=16  feet,  Ans. 
144 

Gangers'  Work. 

To  find  the  contents  of  a  cask  in  gallons. 

RULE. — Add  two -thirds  the  difference  of  the 
head  and  bung  diameters  to  the  head  diameter,  to 
find  the  mean  diameter;  then  multiply  the  product 
of  the  square  of  the  mean  diameter  into  the  length 
by  .0034. 

NOTE.— If  the  staves  are  but  little  curved,  add  six-tenths  instead 
of  two-thirdi. 

How  many  gallons  in  a  cask,  length  40  in.,  head 
diameter  21  in,  and  bung  diameter  30  in.? 


.  .21-f(30— 21X§)=27  in.  mean  diameter. 
...272X40X.0034  =  99.144  gallons. 

Bricklayers'  Work 

Is  sometimes  measured  by  the  perch,  but  more 
frequently  by  the  1000  bricks  laid  in  the  wall. 

The  following  scale  will  give  a  fair  average  for 
estimating  the  quantity  of  brick  required  to  build 
a  given  amount  of  wall : 


58          HOWARD'S  ART  OF  COMPUTATION. 

4^  in.  wall,  per  ft.,  superficial,  (4^  brick)    7  bricks. 

9  «  «  «  (1  brick)  H  " 

13  «  «  «  (l|  brick)  21  « 

18  "  «  (2  bricks)  28  " 

22  "  "  «  (24-  bricks)  35  « 

NOTE.-— For  eac'a  half  brick  added  to  the  thickness  of  the  wall, 
add  eeveu  bricks. 

A  bricklayer's  hod  measuring  1  ft.  4  in.  x  9  in. 
X  9  in.,  equals  1,296  inches  in  capacity,  and  will 
contain  20  bricks. 

A  load  of  mortar  measures  1  cubic  yard,  or  27 
cubic  feet;  requires  1  cubic  yard  of  sand,  and  9 
bushels  of  lime,  and  will  fill  30  hods. 

Plasterers'  Work 

Is  measured  by  the  square  yard,  for  all  plain  work- 
by  the  foot,  superficial,  for  plain  cornices;  and  by 
foot,  lineal,  for  enriched  or  carved  mouldings  in 
cornices. 

Painters'  Work 

Is  computed  by  the  superficial  yard ;  every  part  is 
measured  that  is  painted,  and  an  allowance  is  added 
for  difficult  cornices,  deep  mouldings,  carved  sur- 
faces, iron  railings,  etc.  Charges  are  usually  made 
for  each  coat  of  paint  put  on,  at  a  certain  price  per 
yard  per  coat. 


Howard's    New    Rule  5y 

FOR  COMPUTING  INTEREST 

On  a  Dasis  of  One  Per  Cent  for  all  Rates- 

Interest,  in  the  various  forms  under  which  jt  accrues,  has  so 
large  a  place  in  every  day  business  transactions,  that  a  rapid 
and  accurate  method  of  computing  interest  is  one  of  the  most 
indispensable  items  of  business  knowledge. 

The  one  that  I  present  here,  in  all  respects,  is,  without  ex- 
ception, the  jieicest,  the  easiest  to  learn,  and  use,  the  quickest  and 
most  correct  in  existence,  adapted  to  all  sums,  all  periods,  and  all 
rates  per  cent. 

Some  of  the  reasons  that  suggested  the  construction  of  this  rule 
'will  assist  the  learner  in  its  acquirement  A  child  first  con- 
ceives the  idea  of  ONE  thing,  by  and  by,  it  is  able  to  count  six, 
but  it  is  a  long  time  before  it  apprehends  that  the  six  counted 
is  $ix  ONES. 

THE  UNIT— ^or  one  thing— is  the  idea  of  number  in  its  sim- 
plest form,  it  is  the  basis  of  every  number,  the  primary  base  of 
every  fraction,  the  unit  of  six  months  is  one  month,  the  unit  of 
a  fraction  is  the  reciprocal  of  the  denominator,  thus,  |  is  the  unit 
of  | ;  every  step  from  the  unit,  increases  the  complexity  of 
numbers,  and  consequently  demands  an  increase  of  mental 
power  and  energy  in  dealing  with  them. 

The  most  popular  Interest  rule  is  the  "six per  cent"  method, 
by  this  rule,  on  removing  the  decimal  point  two  places  to  the  left, 
the  interest  on  any  sum  is  shewn  for  ono-sixth  of  a  year  at  six  per 
cent.  The  interest  on  any  sum  is  shown  for  one  year  at 
one  per  cent  by  the  same  act ;  the  latter,  which  retains  the  unit 
of  both  denominations, — unchanged — must  be  the  most  natural 
and  simple  basis  of  calculation,  and  by  consequence,  the  easiest 
to  learn  and  use,  hence  the  following  : — 

Rule, — Multiply  .01  of  the  principal  by  the  given 
time  and  the  product  is  the  interest  at  one  per  cent. — 
Multiply  the  interest  at  one  per  cent  by  tJie  given  rate. 

NOTE  1, — Multiply  by  easy  fractions  of  a  year,  or  month,  and 
the  result  will  be  uniformly  correct,  and  requires  less  than  half 
the  mental  labor  demanded  by  other  methods,  a  little  practice, 
and  careful  study  of  the  details  of  the  following  examples  will 
enable  the  learner  to  select  instantaneously  the  easiest  multipliers, 


60          HOWARD'S  ART  OF  COMPUTATION. 

NOTE  2. — To  multiply  by  .1   remove  the  decimal  point  out 
place  to  thek/J;  by  .01  too  places;  by  .001  three  places. 

What  is  the  interest  on  £1000  for     11     yrs.,  lino 
and  6  days  at  1  per  cent  per  annum  ? 

Yr.    Mo.     Da. 

£10.00  Int.  for       1     "     u 
100.00     "     "      10     "     "  first  lineX  10 
1.00     "     "  16     "      "    X  .1 


£111.00    "     "       11     1     6 


What  is  the  interest  on  $846  for  1  yr.    7    mo.,    12 
da.  at  1  per  cent? 

Yr.    Mo.  Da. 

$8.46  Int.  for  1  "     " 

4.23    "      "     "  6  "  1st  line X£  because  6  mo.  is  £  a  year. 

.705"      "     "  1  "   2d  "     X£         "       1  "     "  I  of  6  mo. 

.282"     "     "  "  12  3d  "    X.4         "     12  da    "  .4  of  1  mo. 

13.677  "     "~~1 7  12. 


What  is  the  int.  on  $427.20  for  2  yr.,  5  mo.,  27  da  at  6  p<T  cent? 
Yr.    Mo.   D  . 


$4.272  int.  for  1 

4.272     "  "  1 

1.424     "  "  <4 

.356     "  «'  " 


at     1     per  cent. 


"     "       "       "     lstlineX-\ 
"     "       "       "     3d      "   Xj 
.3204  "     "       "       "     27        "     "       "       "     4th     "    X.9 


10  6444  "  2,      5,     27         "     1        " 

83.8664  "     **      •«'       "        »«        "      6      "        "  K  37 


What  is  the  int.  on  £124.50  for  1  yr.  4  mo.  12  da. 
at  5  per  cent  per  annum  ? 

£1.245  int.  for  1  yr.  mo.  da. 

.415  "         4      "  1st.  lineXi 

.0415  "         "     12  2d      a    X-l 

"1.7015  1         4     12~ 


£S.5075~Int.  at  5  per  cent. 


INTEREST. 


61 


NOTE.  The  interest  is  found  on  all  sums  at  I  per  cent,  a 
month  by  removing  the  decimal  point  to  the  left,  3  places  for  3 
days,  and  2  places  for  30  days. 

Find  the  interest  on  £143  for  1  mo.  3  da.  at  1  per 
cent  per  month. 
£1.43  int.  for  1  mo. 

.143  «   «     3 'days  1st.  lineX-1 
£1.573     Ans. 


Find  the   int.    on  $216   for  7  mo.  18  da.  at  2  per 
*  cent  per  month. 

$2.16  int.  for  1  mo.  at  1  per  cent. 

15.12    "     "     7    "    «  "     "      «     IstlineXv 
1.296"     "  18  da.  "  "    "      u       "      "  X   -6 


16.416 
$32.832     Int.  for  7  mo.  18  da.  at  2  per  cent.   Ans, 


Find  the  Int.  on  $846.50  for  5  mo.  19  da.  at  1J  per 
cent  per  month. 
$8.465 

"~42i325~  1st  line X 5 

2.822     "       "  X^  because  10  days  is  J  of  a  mo. 
2.5395  "      "  x'.3     "          9     "     t;  .3 

"1T6865 
11.9216  previous  line  Xi 


$59,6081  int.  for  5  mo.  19  da.  at  1J  per  cent   Ans. 

Find  the  Int.  on  £Tl5  for  8  mo.  11  da.  at  .9  of  1 
per  cent  month. 
£7.15     Int.  for  1  mo. 

"57720       "     "     8  " 
1,43       "     "     6  da.  =J.  of  1  month. 
1.192     "     "•    5    "    =i  u  u         " 


59.822 
£53.8398  int.  for  8  mo,  11  da.  at  .9  of  1  per  cent. 


62          HOWARD'S  ART  OF  COMPUTATION. 

Legal  Interest 

is  computed  on  the  basis  of  365  clays  to  the  year. 

The  LEGAL  INTEREST  on  £1,  or  $1,  for  1  da}-,  at  1 
per  cent  per  annum,  is  .0000274,  hence  the  following 

Rule. — Annex  0  to  the  number  of  days,  multiply  by 
274  reversed^  then  annex  0  to  the  number  of  pounds,  or 
dollars,  multiply  by  the  ficfures  in  the  first  product ,  re- 
versed, remove  the  point  four  places  to  the  left,  and  the 
interest  for  the  given  time,  and  principal,  is  shewn  at  1 
per  cent  per  annum.  Multiply  this  by  the  given  rale. 

Find  the  LFXJAL  interest  on  £233  Stg.  for  232  days 
at  7  per  cent  per  annum. 

£2330  2320 

6536  472 

13980  4640 

699  1624 

116  92 

14  6856 


1.4809    int.  at  1  per  cent. 

7  £    s.    d. 


10.S663  int.  at  7  per  cent.    Ans.   10  :  7  :  4 

Find  the  interest  on  £719,,  17,,  9,  for  2  yrs.  7  days,  at  1  per  cent 

£719837  070 

_291_0'J_  472         Note.—  When  the  time  is  in 

143977  140     years  and  clays,  to  the  product  of 

7  oe\  JQ      tne  days  by  472,  prefix  for  1  yr. 

1  **  Q      10,  for  2  yrs.  20  &c.,  and   use 

6  47  o     two  decimals  in  the  princpal. 

" 


__     _ 

14.5^)58  Ans.  £H,,10,,8J. 


INTEREST.  63 

Howard's  New  Rule 

For  Computing  Interest  by  dividing  the  year,  or 
month,  by  the  rate,  may  be  used  in  all  cases  when  the 
figure,  or  figures,  representing  the  rate,  is  the  aliquot 
part  of  a3'ear,or  month,  under  this  rule  the  interest 
can  be  found  in  the  twinkling  of  an  eye,  on  a  million 
examples,  for  three  periods  of  time,  without  altering; 
one  figure  of  the  principal. 

Rule. — Divide  the  year,  or  month,  by  the  rate,  anct 
the  Quotient  is  the  time  in  which  £1  stg.,  or  '  $1 ,  earns  .01 
part  of  itself,  to  find  the  interest  for  the  quotient,  re- 
move the  decimal  point  two  places,  for  .1  that  time, 
three  places,  and  for  ten  times  the  quotient,  one  place  to* 
the  left. 

Find  the  interest  on  $714.50  for  400  days  at  9  per 
cent  per  annum.  Ans.  $71.45. 

Explanation,  360  -4-  9  =  40,  in  40  days  the  dol- 
lar earns  a  cent,  the  interest  is  found  for  40  days  at 
9  percent,  by  removing  the  point  two  places  to  the 
left,  for  10  times  40  days,  by  removing  the  point  one 
place. 

Find  the  interest  on  $125  for  10  days  at  3  per 
cent  per  month.  Ans.  $1.25r 

Explanation  30  da}rs  -f-  3  =  10  days,  in  ten  days, 
one  dollar  earns  one  cent,  at  3  per  cent  per  month, 

Or,  multiply  the  principal  by  the  given  number  of  days 
and  divide  by  the  qoutient  of  360  divided  by  the  given  rate, 
and  remove  the  point  two  places  to  the  left. 
Find  the  interest  on  «£714,,8  for  23  days  @  9  %  per  an. 

714'4  X  23  =  4107.8  Ans.  £4.1078. 

4 

NOTE  1. — When  the  figure  in  the  units1  place  of  the  quotient  is  0, 

divide  by  the  tens  only,  and  remove  the  point  three  places  to  the  left. 

NOTE  2. — Divide  the  number  of  days  in  the  year  by  any  given 

rate,  and  the  quotient  is  the  number  of  dollars  that  will  earn  one 

.  <;ent  in  one  day  at  the  given  rate. 


64 


HOWARD'S  ART  OF  COMPUTATION. 


HOWARD'S  BANK  OF  ENGLAND  RULE 
is  the  handiest  for  computing  Legal  Interest. 

1  X  IjS-r- 1000  =  1  X  4^  -f-  365  X  100  nearly. 
The  difference  is  about  1680;  six  cents  to  be  added 
to  each  $100;    exactly  one  penny  to  each  £7  of 
interest ;  hence  the  following  : 

To  compute  interest  for  any  number  of  days,  365 
days  to  the  year : 

RULE. — Multiply  the  principal  by  1^,  remove  the 
point  three  places  to  the  left,  and  the  interest  will  be 
shown  for  the  following  number  of  days,  and  rates,  to 
find  the  interest  for  any  other  time  or  rate,  increase 
or  diminish : — 


42  days  at    1 
21      *  „         2 
14        „         3 
12        „         3, 

10|       »         4 
Kemove  the 
interest  will  be 
84  days  at    5 
56         „         7^ 
42         „       10 

per  cent. 

M 

99 
;             99 

99 

point  *wo  i 
shown  for 
per  cent. 

99 
99 

7  days  at 

«    » 

4        „ 

3        „ 

9 

w            >j 

)laces  to  the 

35  days  at 

28       „ 

6    per  cent. 

7                  99 

10*       „ 
14         „ 
21 
left,  and  the 

12  per  cent. 
15        „ 

Example :  What  is  the  interest  on  £100  for  fourteen 
days  at  3  per  cent,  per  annum  ? 
100 
10 
5 


.115 

NOTE. — To  multiply  by 
to  itself. 


Ans.  25.  3|d. 

k,  add  T\j  and  J  of  T\y  of  any  number 


OF  THE 


COMPOUND   INTEREST. 


65 


Compound  interest  is  interest  on  the  principal,  and 
also  on  the  interest  added  to  the  principal,  each  time 
it  becomes  due. 

RULE. — Multiply  the  principal  by  the  rate,  setting 
the  product  under,  and  two  decimal  places  to  the  right 
of  the  principal  ;  the  sum  of  principal  and  interest 
will  be  the  amount. 

Or,  find  the  amount  of  «£!,  or  $1,  for  the  given 
time  and  rate,  and  multiply  by  the  given  principal. 

NOTE.— To  avoid  writing  decimals  of  no  value,  begin  at  the  third  decimal, 
adding  in  the  figure  carried,  if  any,  from  the  right  hand  figures. 

Find  the  amount  of  £864  10s.  Od.  for  six  years  at  8%. 
Ans.  £1371  17s.  0|d. 

School  Book  Method,  184  Figures. 
864]5 
8 

69160 
8615 
933.060 

8_ 

746928 

_93366 

1008.3528" 


Howard's  Method, 
74  Figures. 


864.5 
69.16 


74.693 
1008.353 

80.668 
1089.021 

87.122 
1176.143 

94.091 

1270.234 

101.619 

11371.853 


80668224 

10083528 
1089.021024 

8_ 

8712168192 

1089021024 

1176.14270592 
8_ 

940914164736 
117614270592  _ 
1270.2341223936 

8 

1016l872979148a 

1270.2341223936 
£1371.85285.2185088 


66          HOWARD'S  ART  OF  COMPUTATION. 

To  repay  a  loan,  principal  and  compound  inter- 
est in  a  given  number  of  equal  qnnual  payments. 

RULE.  —  Multiply  the  amount  of  one  pound,  or 
one  dollar  for  the  given  time  and  rate,  by  the  inter- 
est for  one  year,  and  divide  the  product  by  the  com- 
pound interest  on  a  pound,  or  dollar,  for  the  given 
time  and  rate 

EXAMPLE.  —  What  must  the  be  one  of  six  equal 
annual  payments  to  discharge  a  loan  of  £864,,  10,, 

for  six  years  at  8  per  cent. 
1.5869x69.16 


.5869 

NOTE  1.—  Persons  having  frequent  occasion  to  com- 
pute compound  interest  may  save  time  and  labor  by  the 
use  of  a  table  showing  the  amount  of  one  pound,  or  one 
dollar,  for  a  series  of  years,  or  <5ther  stated  periods  ;  the 
amount  of  one  pound,  or  one  dollar,  for  the  given  time 
and  rate,  multiplied  by  the  given  number  of  pounds,  or 
dollars,  will  be  the  amount  sought. 

NOTE  2.  —  To  prove  interest,  divide  the  computed  in- 
terest by  the  interest  for  one  day,  and  the  quotient  should 
be  the  number  of  days  in  the  example  ;  or  divide  by  the 
interest  for  one  month,  and  the  quotient  should  be  the 
number  of  months. 

DISCOUNT. 

Discount,  being  of  the  same  nature  as  interest, 
is,  strictly  speaking,  the  use  of  money  before  it  is 
due.  The  term  is  also  applied  to  a  deduction  of 
so  much  per  cent,  from  the  face  of  a  bill,  or  the 
deducting  of  interest  from  the  face  of  a  note  before 
any  interest  has  accrued.  Banks  generally  include 


DISCOUNT.  67 

in  their  reckoning  both  the  day  when  the  note  is 
discounted  and  the  day  on  which  the  time  specified 
in  it  expires,  which,  with  three  days  of  grace, 
makes  the  time  for  which  discount  is  taken  four 
<ia}rs  more  than  the  time  specified  in  the  note. 
True  Discount  differs  Irom  Bank  Discount,  that  is, 
the  true  discount  on  a  debt  of  109  dollars  due  a 
year  hence  would  be  9  dollars,  the  legal  interest 
being  at  the  rate  of  9  per  cent.,  and  the  present 
of  the  note  is  100  dollars. 


In  calculating  interest  the  sum  on  which  interest 
is  to  be  paid  is  known,  but  in  computing  discount 
we  have  to  find  what  sum  must  be  placed  at  inter- 
est, so  that  the  sum,  together  with  its  interest,  will 
amount  to  the  given  principal  ;  the  sum  thus  found 
is  called  the  "  Present  worth." 

To  find  the  present  worth  of  any  sum,  and  the  dis- 
count for  any  time  at  any  rate  per  cent 

RULE.  —  Divide  the  given  sum  by  the  amount  of 
$1  for  the  given  time  and  rate,  and  the  quotient 
will  be  the  present  worth,  and  the  remainder  will 
l>e  the  discount. 

EXAMPLE  1.  —  Find  the  present  worth  of  a  note 
for  228  dollars,  due  2  years  from  date  at  7  per  cent. 

Ans. 


2.  Find  the  bank  discount  on  .a  note  for  £1200, 
for  60  days  at  6  per  cent. 


68          HOWARD'S  ART  OF  COMPUTATION. 

60  -|-  4  —  64  days  time  for  which  discount  must 
be  reckoned.  £  of  64  —  lOf  X  1200  =  12  800. 

,    Ans.  12.80. 

Merchants  are  in  the  habit  of  deducting  a  certain 
percentage  from  invoices  of  goods  sold.  This  is- 
reckoned  in  the  same  manner  as  interest. 

A  bill  of  goods  is  bought,  amounting  to  960  dol- 
lars at  a  year's  credit,  the  merchant  offers  to  de- 
duct 10$  for  ready  cash,  what  amount  is  to  be 
deducted  ? 

9.60  x  10  =  $96.00,  Ans. 

By  discounting  the  face  of  bills,  a  loss  may  be 
sustained  without  suspecting  it;  this  arises  from 
the  fact  that  the  discount  is  not  only  made  on  the 
first  cost  of  the  goods,  but  also  on  the  profits ;  for 
instance,  if  a  profit  of  30%  be  made  on  any  article 
of  merchandise,  and  the  10$  be  deducted,  the  gain 
at  first  sight  would  appear  to  be  20$,  but  is  in 
reality  only  11%.  If  a  profit  of  60$  be  added  to 
the  first  cost,  and  then  a  discount  made  of  45$,  the 
apparent  profit  would  be  15$;  instead  of  this,  an 
actual  loss  is  made  of  12$,  as  will  be  seen  by  the 
following  examples : 

Example  1.  Example  2. 

Cost  of  goods,  $100         Cost,  $100 

Add  30$  profit,  30         Profit  60$,  60 


Selling  price,                   130  Selling  price,  160 

Deduct  10$  discount,     13  Discount  45$,  72 

Cash  price,                    $117  Cash  price,  $88 

Gain  17$.  Loss  .12$. 


DISCOUNT.  69 

The  net  amt.  of  a  bill,  less  10  per  cent  discount,  will  be  shewn 
by  multiplying  by  9,  and  removing  the  decimal  point  one  place 
to  the  left.  Example.  £100X9=^90.0 

To  find  the  net.  amt.  less  discount  at 

5  per  cent  X  9^.         30  per  cent  X7.         50  per  cent  X5. 

15    "      "    X8£.         35    «      "     X6£.       55     "      "     X4£. 

20    "      "    X8.  40    "      "     X6.         60     "      "    X4. 

25    "      "    X7£          45    «      "     X5i-       70    "      "     X3. 

-.    and  remove  the  point  1  place  to  the  left. 


EXCHANGE. 

EXCHANGE  is  the  giving  or  receiving  of  any    sum 
in  one  kind  of  money  for  its  value  in  another. 

EXAMPLE  1.     Find  the  value  of  gold,    the  price 
of  greenbacks  being  75  cents  Ans. 

100         4 

Process—         =  —  =  1.38J 

75          3 


2.     Find    the  value    of   currency,   the    price   of 
gold  being  133J,  Ans.  75  cents, 

100     .    3 
Process  —         -  =  —  =.75 


___ 

$500  in  gold  at  8  per  cent,  premium  will  buy  how 
much  currency? 

$500xl.08==$54G 

$500  in  currency  will  buy  how  much    gold    at    8 
per  cent  premium? 

5QO~108=$462.96. 

$1000  in  gold  is    worth  how    much     currency   at 
80  cents? 


70          HOWARD'S  ART  OF  COMPUTATION. 

What  is  the  face  value  of  a  bill  of  Exchange  cost- 
ing £1000.     Commission  f  per  cent  ? 
£1000-^1.0075=£992.55 


What  is  the  cost  of  a  bill  of  Exchange  for  $1  000 
Premium  f  per  cent. 

$1000X1. 


Find  the  par  value  of  £473  „  5  „  9  St'g.  in  Amer- 
ican gold  coin. 

'  £473.2875  X  4.8665=$2303.25. 

473.2875 

Note.  To  avoid  encumbering  the  operation  with  56.684 
valueless  decimals,  reverse  the  multiplier,  and  begin  1893.150 
each  line  of  the  partial  products  with  the  product  of  378.630 
the  multiplying  figure  and  the  figure  directly  above  28.397 
it,  adding  what  otherwise  would  have  been  carried.  2839 
The  par  value  of  £1  st'g  is  fixed  by  act  of  Con-  .237 

gress  1873,  at  $4.8665.  ~2303.254 

BRITISH  MONEY. 

Howard's  new  rules  for  INTEREST,  EQUATION  OF 
PAYMENTS,  &c.,  may  be  used  with  equal  facility  in 
dealing  with  British  and  other  foreign  money. 

The  British  people  would  simplify  all  their  mon- 
etary operations,  and  save  millions  every  year  in 
labor  alone,  by  adopting  the  decimal  system  of 
coinage.  The  cost  and  temporary  inconvenience 
incident  to  the  change  would  be  trifling,  almost 
nil,  in  view  of  the  advantage  to  be  gained.  The 
pound,  the  florin,  the  shilling  and  the  sixpence 
might  be  retained.  Make  the  smallest  coin,  the 
farthing,  equal  to  the  T17Vo  of  a  pound,  and  the 
thing  is  done. 


BRITISH    MONEY.  71 

NOTE.—  By  carefully  observing  and  practicing  the  following  in- 
structions, the  converting  of  shillings,  pence  and  farthings  into 
decimals  of  a  pound,  and  vice  versa,  will  become  a  purely  mental 
and  instantaneous  operation. 

1.  For   every   two   shillings,  or  florin,  write  .1, 
because  two  shillings  is  ^  of  a  pound  stg. 

2.  For  every  1  shilling,  write  .05,  because  one 
shilling  is  T5^  of  a  florin,  or  ^fa  of  a  pound  stg. 

3.  For  every  ninepence,  write  .0375,  because 
ninepenee  is  ^So  of  a  pound  stg. 

4.  For  every  sixpence,  write  .025,  because  six- 
pence is  100  of  a  florin  or  ^  of  a  pound  stg. 

5.  For  every  threepence,  write  .0125,  because 
threepence  is  j^w  of  a  pound  stg. 

6.  For  the  farthings,  write  the  product  of  .00104 
multiplied  by  the  number  of  farthings. 


£      s. 
1     =  20 
.1    =  2 
.05   =  1 
.0375  =  | 
.025  =  i 
.0125  =  j 

d.       far. 
=  240    =  960 
=  24    =96 
=  12    =48 
=   9    =36 
=   6    =24 
=   3    =12 

.00104=  ^ 

=     4"       ==     1 

£  19,,2 

=  19.1 

£  27,,12,,6  =27.625 

19,,3 

=  19.15 

19,,19,,2i=19.96 

19,,5 

=  19.25 

19,,18,,0f=19.903 

19,,19 

=  19.95 

19,,16,,lf=19.807 

19,,18 

=  19.9 

24,,  1,,H=24.056 

The  learner  may  extend  the  exercises  indefinite- 
ly, the  essentials  to  remember  are  — 


72          HOWARD'S  ART  OF  COMPUTATION. 

1st.  Each  unit  of  the  first  figure  to  the  right  of 
the  decimal  stands  for  two  shillings. 

2d.  Each  5  in  the  second  figure  to  the  right  of 
the  decimal,  stands  for  one  shilling. 

3d.  j^ach  unit  above  or  below  c  in  the  second 
figure,  stands  for  2£  pence. 

4th.  Each  unit  of  the  third  figure  to  the  right 
of  the  decimal,  stands  for  1  farthing. 

NOTE.— The  exact  value  of  each  unit  in  the  sec- 
ond figure  to  the  right  of  the  decimal  is  2T\-  ol  a 
penny,  and  of  each  unit  in  the  third  figure  to  the 
right  ol  the  decimal,  -f-^  of  a  penny,  the  difference 
of  the  assumed  and  the  real  value  is  too  trifling  to 
affect  any  actual  business  operation.  The^/forms, 
shillings,  ninepence,  sixpence  and  threepence  are 
decimally  expressed  absolutely  correct. 

PERCENTAGE. 

The  following  examples  embrace  most  of  the 
conditions  under  which  percentage  occurs  in  busi- 
ness, and  the  mode  of  solution  in  each  case  applies 
to  all  similar  examples. 

How  many  of  500  sheep  will  be  left,  if  20  per  cent, 
of  them  are-  sold  ? 

600X.20=100.  500—100=400  sheep. 

What  per  cent  of  300  is  75?     75-f-300=25  $>  ct. 

Of  what  number  is  48,  8  $>  ct.?     48-^.08=600. 

Sold  a  horse  for  £60,  made  25  $>  ct.,  what  did  it 
cost?  1+.25=H{}=S  «  I  a*0=i£48 


PERCENTAGE.  73 

Sold  a  horse  for  §40,  lost  20  $>  ct.     What    did   it 

cost? 


The  population  of  a  village  increased    from    900 
to  1200,  at  what  rate  per  cent,  did  it  increase  ? 
1200-^-900=1.33^— 1=33  J  per  cent. 

The  sales  of  a  firm  fell  off  from  £12000  to  £9000, 
what  was  the  rate  per  cent  of  decline? 
9000-^-12000=.75.     1  — .75=25  per  cent. 

Bought  a  horse  for  $80,  sold  it  for  $105.  What 
per  cent  profit? 

105-^-80=1.311  —  1=311  per  cent. 

Bought  a  piano  for  $300,  sold  it  for  $250.  What 
per  cent,  loss? 

300—250   -f-  300  =  .16|  per  cent. 

Bought  a  horse  for  $40.  What  must  it  be  sold 
for  to  gain  20  per  cent? 

40  X- 20=8+40=48  dollars. 

A  horse  was  sold  for  $24;  the  rate  per  cent  profit 
was  the  same  as  the  number  of  dollars  it  cost.  What 
was  the  cost,  and  what  the  gain  per  cent? 

Cost  $20.     202=400X.01=4.     Profit  $4,  or 

Vvf  the  profit  is  .1  the  cost.       Vof  4=2x10=20 
Cost  $20.     Profit  2Qpcr  cent. 

How  many  dollars  will  earn  1  cent  a  day  at  9  per 
cent  per  annum? 

360-7-9=40.     AnsMO. 

How  many  dollars  will  earn  1  cent  a    day    at    1^ 
per  cent  per  month? 
30-f-lJ=24.       ' 


74          HOWARD'S  ART  OF  COMPUTATION. 

STOCKS  AND  BONDS, 

Stocks  and  bonds  are  quoted  in  New  York  by  so 
much  on  the  hundred,  premium  or  discount;  in 
Philadelphia  at  their  actual  price.  That  is,  if  the 
par  value  of  a  stock  is  $50,  and  it  is  6%  above  par, 
the  New  York  quotation  would  be  106,  the  Phila- 
delphia quotation  53. 

When  the  premium  is  known,  the  par  value  plus 
the  premium  equals  the  market  value.  When  at  a 
discount,  the  par  value  minus  the  discount  equals 
the  market  value. 

To  find  to  what  rate  of  interest  a  given  dividend  cor- 
responds. 

RULE. — Divide  the  rate  per  unit  of  dividend  by 
1  plus  or  minus  the  rate  per  cent.,  premium  or  dis- 
count, according  as  the  stocks  are  above  or  below 
par. 

What  per  cent  will  be  gained  by  investing  in  8  pet 
cent  stock,  at  20  per  cent  premium? 

120  |  800=;6§2HT  cent. 

What  per  cent  will  be  gained  by  investing  in  6  per 
€ent  stock  at  10  per  cent  discount. 

100  — 10=90.       90  |  600=63  per  cent. 
To  find  at  what  price  stock  pay  ing  a  given  rate  per 
cent,  dividend  can  be  purchased,  so  that  the  money  in- 
vested shall  produce  a  given  rate  of  interest. 

Hi .- I.E.— Divide  the  rate  per  unit  of  dividend  by 
the  rate  per  unit  of  interest. 

What  must  be  paid  for  stock  paying  6  per  cent  divi- 
dend t  in  order  to  realize  on  the  investment  8  per  cent? 
8  |  600=75. 


EQUATION  or  PAYMENTS.  7  a 

HOWARD'S 
GhOXJDHiT    DE^-CnLiE. 

FOR 

EQUATION     OF     PAYMENTS, 

AVERAGING  ACCOUNTS  and  PARTIAL  PAYMENTS,, 
is  so  called,  not  only  because  it  is  absolutely  cor- 
rect, and  consequently  equally  just  to  both  Debtor 
and  Creditor,  but  also  because  it  is  exceedingly  sim- 
ple, and  easy  to  learn  and  use. 

The  methods  hitherto  in  use  are  intricate,  tedious, 
and  perplexing,  and  more  or  less  inaccurate:  the  PRO- 
DUCT methods  requiring,  with  each  item,  the  find- 
ing the  number  of  days  between  two  dates,  and  the 
use  of  difficult  multipliers. 

The  INTEREST  methods  introduce  a  superfluous 
element,  and  needlessly  increase  the  complexity  of 
the  operation.  Interest,  really  has  nothing  to  do 
with  finding  when  a  balance  is  due. 

The  object  sought  is  a  certain  date,  HOWARD'S 
GOLDEN  RULE  seeks  and  finds  this — and  this  only — 
directly,  accurately  and  easily.  By  its  use  the 
CASH  BALANCE  of  the  most  complex  Dr.  and  Cr. 
accounts  may  be  easily  found,  without  reference  to 
interest,  except  where  it  properly  belongs;  viz,— 
on  the  balance. 

The  novel  and  special    excellence    of   this    rule 
consists  in  multiplying  by  months,and  easy  fractions 
of  a  month,  and  also  in  the  simple  and   natural   ar- 
rangement of  the  parts  of  the    problem,    the    dates. 
themselves  representing  the  multipliers. 

Experienced  Accountants  say,  "  it  very  much 
lessens  the  drydgery  of  the  counting  house." 


76          HOWARD'S  ART  OF  COMPUTATION. 


EQUATION  OF  PAYMENTS  is  the  process  of  find- 
ing the  EQUATED  TIME,  or  the  date  when  the  sum 
of  several  debts  due  at  different  times  may  be  paid 
and  includes, — 

Bills  bought  ou  unequal  time  on  the  same  date. 

Bills  bought  oil  equal  time  on  different  dates. 

Bills  bought  011  unequal  time   on  different  dates,  and  MONTHLY 

STATEMENTS. 

AVERAGING  ACCOUNTS  is  the  process  of  finding 
the  date  on  which  the  BALANCE  is  due,  and  applies 
to  all  Dr.  and  Cr.  accounts. 

PARTIAL  PAYMENTS  are  parts  of  a  debt  paid  at 
different  times ;  usually  written  on  the  back  of 
notes  and  other  interest  bearing  obligations,  and 
called  indorsements.  The  term  also  includes  pay- 
ments made  on  account  of  a  debt  before  it  is  due. 

TERM  OF  CREDIT  is  the  time  to  elapse  before  a 
Dill  becomes  due. 

The  AVERAGE  TERM  of  credit  is  the  time  at  the 
end  of  which  the  sum  of  several  debts  due  at  differ- 
ent dates  may  be  paid  at  once. 

EQUATED  TERM  is  the  average  time  for  which  in- 
terest is  due  on  an  account,  or  balance,  and  is  al- 
ways reckoned  from  the  zero  date. 

Interest  is  reckoned  on  accounts,  and  balances  from  the 
date  on  which  they  are  due. 

AN  ACCOUNT  is  a  statement  of  business  transac- 
tions between  Debtor  and  Creditor. 

A  BALANCE  is  the  difference  of  two  sides  of  an  ac- 
count. 

A  CASH  BALANCE  is  the  same,  with  the  interest 
due. 

THE  ZERO  DATE  is  the  date, — or  starting  point, — • 
from  which  all  the  other  dates  are  reckoned,  in  this 
rule  it  is  always  the  beginning — or  starting  point — of 
the  month  in  which  the  first  debt  in  the  acct.  occurs. 


EQUATION  OF  PAYMENTS.  77 

BILLS  BOUGHT  ON  UNEQUAL  TIME  ON  THE  SAME  DATE. 


1878, 

Jan. 

1st, 

Bought 

goods  on 

8 

mos. 

£100 

tc 

a 

" 

" 

u    a 

b 

7 

i  4 

100 
100 

On  what  date  may  the  whole  £300  be  paid? 


Term  of 
CrMo.. 

8 
6 
7 


Jan.  1. 


100X8=800 
100X6=600 
100X1=700 


300      )2100(7mo.fr.Jan.l,orAug.l, 

Under  the  terms  of  this  transaction  the  Debtor  is  entitled  to 
the  use  of 

1st,  £100  for  8  months,  =8  times  loo   or-£Soo  for    i  mo. 
2d,  100    "    6        "       =6      "      100   "     600     "     i     " 
3d.  100    "7        "       =7     "      100   "     Too    "     i     "   ' 
a  credit  equal  to  €2100  for  1  month  ;  this  will  evidently   entitle 
the  debtor  to  the  use  of  £300  for  as   many   months    as    300   is 
contained  in  2100. 

The  product  of  any  number  of  pounds  multiplied 
by  any  number  of  months,  and  fractions  of  a  month, 
a  Debtor  is  entitled  to  use  them,  is  the  number  of 
pounds  he  is  entitled  to  use  for  1  month  under  the 
same  terms,  hence  the  following  ; — 

Rule. — Multiply  each  debt  by  its  term  of  credit^  divide 
the  sum  of  the  products  by  the  sum  of  the  debts,  and  the 
quotient  is  the  equated  term. 

First  study  this  very  simple  example  thoroughly, 
make  yourself  -familiar  with  each  operation,  the 
reason  for  its  use,  and  the  causes  of  the  results,  and 
you  will  then  have  no  difficulty  in  comprehending 
the  most  complex  Debtor  and  Creditor  accounts. 


78          HOWARD'S  ART  OF  COMPUTATION. 

BILLS  BOUGHT  OX  EQUAL  TIME  AT  DIFFERENT  DATES. 

Required  the  equated  time  of  paying  the  follow- 
ing bills  each  bought  on  8  months  credit. 


1878 

No  of  months,      June      0—  Zero  date 

from  zero  date  Tune 

9  X180  X 

.3  —       54 

1 

July 

15 

84  X 

11  l     84 
X2  j     42 

3 

Sept. 

14 

240X3. 

Qi        (720 

8H  n 

4 

Oct.    10 

96X 

4J=j»i« 

£600 

)1428(2.38 

$ 

mo.  da.  yr.  mo.  da.  11.4 

Equated  term       2,  1 1  )  after  78,    6,    0  zero  date. 
Plus  term  of  Cr.  8,     0  j  :=__    10,  11, 
Equated  time  ~79,    4,  if,  or  April  llth,  1879. 

Rule, — Multiply  each  debt  by  the  time — in  months 
and  fractions  of  a  month, —  between  its  occurrence  and 
the  zero  date,  divide  the  sum  of  the  products,  by  the  sum 
of  the  debts  i  and  the  quotient  is  the  equated  term —  in 
months  and  hundredths  of  a  month, —  counting  from  the 
zero  date,  add  the  term  of  credit^  and  the  sum  is  the 
equated  time. 

NOTE  1.  To  reduce  hundredths  of  months  to  days,  multiply 
by  3,  and  point  off  the  right  hand  figure,  when  the  right  hand 
figure  in  the  product  is  5  or  more  add  1  day,  otherwise  disre- 
gard it 

NOTE  2,  When  the  figures  representing  the  day  of  the 
month  are  multiples  of  3,  such  as  the  3d,  9th,  27th,  &c.  &c., 
multiply  by  tenths,  because  3  days  is  .1  of  a  month  ;  when  they 
are  not  multiples  of  3  then  multiply  by  the  simplest  fraction, 
or  fractions  of  a  month.  In  the  above  example,  Sept.  14th,. 
3  months  14  days  from  zero  date,  we  multiply  by  3.3  \  ,  3 
months,  plus  9  days,  plus  5  days.  Facility  in  selecting  the 
simplest  fractions  for  multipliers  is  easily  acquired  by  practice. . 
Et  ssb 


EQUATION  OP  PAYMENTS. 


79 


BILLS  BOUGHT  ON  UNEQUAL  TIME  AT  DIFFERENT  DATES. 

Required  the  equated  time  of  paying  the  follow- 
ing bills  of  goods.    • 

Term  of  -i      A 

Cr.Moa.    April      0 


6f      " 

10 

To  Mdse. 

£310X6J 

— 

[1SGO 
103 

o 

1  May 

21 

u         i' 

468X3.7 

= 

1404 
328 

4 

2  June 

1 

«          u 

520X6^ 

- 

3120 
IT 

4500 

3 

3  July    8 

u          « 

750X6.  1£ 

75 
125 

MO.    Da.                   OAtQ 

Zero  date            4       0                 20*8 

)11532{5.63 

Equat 

id  term     5     1  9 

3 

18.9 

Rule — Multiply  each  debt  by  the  term  of  credit,  plut 
the  time  "between  the  date  of  the  transaction  and  the  sere 
date-,  divide  the  sum  of  the  products  by  the  sum  of  the 
debts,  and  the  quotient  is  the  equated  term. 

The  figures,  on  the  extreme  left  represent  the 
terms  of  credit;  the  figures  on  the  left  of  the 
month  represent  the  number  cf  months  from  the 
zero  date,  these  together  with  the  day  of  the  month 
are  the  multipliers. 

moe.  meg.  da- 

1st  item  6  Cr.  plus   0,  10  from  0  datc=6J  mos. 
2d      "      2     "      "      1,  21     "      "     "    =3,7     " 
3d     "      4     "      "      2,'     1     "    "       "   =6^   <c 
4th    "      3     "     "     '  8,     8     "    u       "    =6.14  " 

Note.— -The  use- of  the  beginning  of  the  month,  instead  of 
the  dafe  of  the  first  transaction  for  the  starting  point,  makes 
no  difference  in  the  ultimate  result,  and  avoids  the  continual 
lalwr  of  finding  on  each  item,  the  time  between  two  dates, 
each  date  as  written,  itself  re  presenting  the  time, 


80 


HOWARD  S    ART    OF    COMPUTATION. 


MONTHLY    STATEMENTS. 

Find  the  equated  time  for  paying  he  following  acct* 

1878 


Jan. 

1  To  Goods 

$660.00  X      A 

=     -22 

3 

u           u 

841.  "  X  -1 

=.       84 

4 

U                .i 

730.  "  X  -1^ 

_.\n 

=  J24 

6 

(  i           t  ; 

786.  "  X      i 

rzz      1  o  1 

6 

u           u 

815.  "X      i 

=     1G3 

8 

a           a 

612.  u  X  -If 

(     61 

=  j  102 

10 

u          u 

312.  "X       -\ 

=     104 

11 

U           u 

215.25  X  ii 

_    J    43 
~    1    36 

15 

u           t  < 

118.  "X       } 

=.        59 

16 

t< 

30.  -  X   H 

__    i  10 
—          6 

19 

u           u 

86.  u  X  -3  J 

—        26 

~    j    29 

20 

u          u 

66.  "X      § 

=        44 

23 

u          u 

48.  «x  -6J 

(    29 

=   J      8 

27 

a          u 

100.  "  X  .9 

~        90 

28 

u           « 

27.  <4  X  .6J 

(    16 

—  •    i      9 

30 

u           u 

48.75X1 

—        49 

5495. 

)1218( 

.22 

3 

Equated  tune  Jan  7th. 


, — Multiply  each  debt  by  the  time  between  its  occur- 
rence and  the  zero  date^  divide  the  sum  of  the  products  by 
the  sum  of  the  debts, andthe  quotient  is  the  equated  term 
This  example  is  extended  for  the  purpose  of  intro- 
ducing every  possible  fraction,  of  a  month,  the  selec- 
tion of  the  simplest  fractions  for  multipliers  will  be- 
come the  work  of  an  instant  by  practice. 

XOTE.— Omit  the  cents  when  under  fifty,  add  one  dollar  when  they 
are  fiity  or  more.    If  English  money,  use  one  decimal. 


AVERAGING    ACCOUNTS 


81 


Find  the  equated  time  of  paying  the  balance  of  the  following  accts 
1878  Dr.  1S7S  Cr. 


Mar. 

lApr 

3.  May 

Bal 

1877 

0 
15 

3 

1( 

an 

3mos 
4    " 

6    " 

•  OOOX3*=:J  1|00 
700X*5.1=  j  3500 

Mar 
2  May 

4  July 
5  Aug 

0 

lo 

1 

r, 

P»y  Cash 

300X2*  =  }    JOO 
400X4^0=  j  16™ 

2300                14003 
1200                 5063 

1200                   51163 

1100               )8940(8.13 
3 

ce  due  Nov.4tli,  8  mos.  4  days  after  zero  date. 

Cr.            .                  1877 

Dr. 

June 
Uuly 

3  Dec. 

1878 
a  Mar 

Zero 
Minu 

0 
4 

IS 
5 

da1 

s 

By  Note 
"  Md'c 

,e       77 

(    158 
L58xl.lKo=l      16 
(       5 
228X6.6     —  \  1368 
i   137 

June 

5  Nov. 
1878 
8  Feb. 

S.093 
2.7    8 

0 
20 

16 
26 

in 

To  Goods 

C329 

(760 
152xW/5=<  51 
C  30 

(   18 

$>(>                    5809 
2474 

1248                 2474 

836 

412)3335 
,    6,    0 
8,    3 

412 
o.  3da  before  zero  date. 

76,    9,  27    Balance  due  Sept.  27th,  1876. 

In  this  example  the  balance  of  the  products  is  on 
the  smaller  side  of  the  account;  when  this  happens 
the  eqated  term  is  deducted  from  the  zero  date  to 
find  the  equated  time. 

The  credit  side  has  the  ADVANTAGE  of  the  use  of 
the  equivalent  of  £"3335  for  one  month,  then  the 
other  side  is  entitled  to  interest  on  the  balance  for 
as  many  months  as  412  is  contained  in  3335. 

Rule. — Multiply  each  item  "by  the  time  between  its  occur- 
rence and  the  zero  date,  added  to  the  term  of  credit — if 
any —  divide  the  balance  of  the  products  by  the  balance  of 
the  account  and  the  quotient  is  the  equated  term. 

Note, — This  rule  applies  to  Partial  Payments  and  all  Dr.  and 
Cr.  accounts. 

R  JI4.          E  24J 


82          HOWARD'S  ART  or  COMPUTATION 

PAUTI A  L    PA  Y  M  EXTS. 

A  note  is  made  March  15th,  1878  for        €T20. 

Endorsed  April  3d,  pd.  on  account  £170 
May  20th,  "  "  "  .  246 
June  18th  "  "  "  87 

from  what  date  must  interest  be  computed   on    the 

balance. 


Aprt 

8|170Xl.l  = 

170 
17 

Mar. 

15 

720  Xi—  360 

2  May 

20 

245  X2§    = 

490 
164 

502 

3  Ju. 

17 

S7X3Ak=\*ll 

218 

I     14 

502                         1151 

360 

2 1 «)  7  9 1(3 .  C  2  Sraos.  19  da.  before  0  date. 

Zero  date      78,  3,     0 
Minus  3,  19 

Equated  time  7, 11, 11         Nov.  llth,  1877. 

PAYM'TS  MADE  OX  ACC'T  OF    A    DEBT    BEFORE  IT  IS    DUE 

[878 Cr.          1878 Dr. 

March  l  By  Cash  $306i  Jan.  1st,  to  goods  on  6  mos.  $1500 

May      1  "      ki        4001 

On  what  date  is  the  balance  $800  due. 


Jan 


1500X6—9000 
2200 


2  March  l!  300X2=  600 


800)6800 
~ 


4  May 


400X^=1600 


700 


2200 


Ans.  8J  mos.  after  Jan.  1st,         Sept.  15th. 

Explanation,/ — under  the  terms  of  this  transaction  the  debtor  is 
entitled  to  the  use  of  $1500  for  6  months,  equal  to  6  time*  1509 
or  $9000  for  1  month  on  paying 
$300  in  2  moe.  the  use  of  which  for  that  time  is=$GOO  for  1  mo. 

400    *«  4    "         "       '•    "        "         "       "       "       "=  1600    •«    1    " 
he  has  used  the  equivalent  of  $2900  for  1  month  and  is   conse- 
quently entitled  to  the  use  of  the  balance   for   a  time   equal   to 
the  use  of  $6800  for  one  month. 

T.  2bq       jR.  /or 


CASH  BALANCES.  83 

The  Creditor  is  entitled  to  interest  on  the  Balance 
from  the  date  on  which  it  is  due,  to  the  date  of  set- 
tlement. The  Debtor  is  entitled  to  discount  off  the 
Balance  for  the  time  he  pays  it  before  it  is  due. 

Find  the  Cash  Balance  on  each  of  the  four  preceding  acc'ts. 
1st,— £1100  clue  Nov.  4th,  settled  Aug.  22d,  int.  at  6  per  cent. 


mo.  da. 

Balance  due              11,   4  Am'tof£K 

Date  of  settlement,   8,  22  mo.    da.  (1.012 

Difference,               ~2,  12  for  2      12  ) 


Cash  Balance 
1100=£1086.95.  Ans. 


2d,— .£412  due  27  |  0  |  76,  date  of  settlement  27  |  7  |  78.^^,. 

Yr.       Mo.       Da. 

78       7       27 

76       9        27  Yr.  Mo. 

~1     10  Int.  for  1,   10,     45.32+  4l2=f457.32.  Ans. 


3d,— £218  due  11  |  11  |  77,  dale  of  settlement  5  |  9  |  78  J"^68^ 

78      9       5 

77     11      11  MO.  Da. 

9      24         Int.  for  9    24,    10.68+ 2l8=£228-68.      Ans. 


4th,— $800  due  78  |  9  |  15  |  date  of  settlement  78  |  12  |  197*£ct 

Mo.    Da. 

12     19 

9     15  MO.  Da. 

3      4        Int.  for  3,    4,    14.62+S800=$S14.G2.    Ans. 


TO    FIND   THE   DIFFERENCE    OF    TIiME   BETWEEN    TWO   DATES. 

Rule.  Subtract  the  earlier  J¥om  the  latter  date. 

Example. — For  what  time  must  interest  be  charged  on  a  debt 
due  the  first  of  May,  1873,  and  settled  on  the  ninth  of  March, 
1875. 

Process,         75  :  3  :  9 
73  :  5  :  1 

1  :  10  :  8  Ans.  1  yr.  10  mo.  8  days. 

NOTE.— To  compute  on  a  basts  of  365  days  to  the  year,  add  one  day  for 
•each  month  of  31  days  ;  deduct  2  days  in  the  common  year,  and  one  day 
in  leap  year,  for  February. 


HOWARD'S  ART  OF  COMPUTATION. 

33  DE  CALCUL  POUR  L'ESPACE  DE  TREKTE 
SIECLES. 

BEGLE. — Des  deux  derniers  chiffres  de  1'an,  rejetez 
tous  ]es  sept,  tout  en  retenant  le  restant;  divisez 
les  deux  derniers  chiffres  de  Tan  par  quatre,  re- 
tenant  le  quotient,  suns  tenir  coinpte  du  restant, 
s'il  y  en  a ; — puis  prcnez  le  jour  du  mois,  ensuite 
le  chiffre  donne  pour  le  mois,  et  finalement  celui 
pour  le  siccle.  Ayez  toujours  soin  de  rejeter  les 
sept  oil  il  y  en  a. 

Le  chiffre  1  (un)  restant  represente  le  premier ; 
2,  le  second ;  &c.,  et  0  (zero)  le  dernier  jour  de  la 
semaine. 

TABLE  DES  CHIFFRES  POUR  LES  MOIS. 

1,  Septembre  et  Dec.        3,  Jan.  et  Oct.        5,  Aotit. 

2,  Avril  et  Juillet.  4,  Mai.  6,  Fev.,  Mars,  Nov.    0,  Juin. 
NOTA. — Dans  l'anc£e  bissextile  le  chiffre  pour  Janvier  est  2,  et  celui 

pour  Fevrier  5. 

TABLE  DES  CHIFFRES  POUR  LES  SIECLES. 

1,  est  le  chiffro  pour  les  2e"me,  9'dme,  et  166me,  sidcles.  [sie~cleg. 

ler,  Seme,  15dme,  18eme,  22eme,  266me,  3Ueme, 
76me,  14eine  siecles.  [sidclea. 

66me,    13dmef    17eme,    ?16me,   2ff6me,   296me, 
6eme,  12eme,  20dme,  24eme,  28dme,  siecles. 
46me,  lldme  sidcles. 
3eme,  lOeme,  196me,  23dme,  27eme,  sleclea. 

EXEMPLE. — Quel  futle  jour  de  la  semaine  au  31 
Aoiit,  1873  ?  R^ponse,  Dimanche. 

Procedd — 

Deux  derniers  chiffres  de  Tan,  73 — 70=3 
Quotient  de  73  di vise  par  quatre,  18  +  3 — 21  =  0 
Jour  du  mois,  31 — 28=3 

Chiffre  pour  le  mois,  5  +  3 — 7=1 

Apres  avoir  rejetcS  tous  les  sept  il  reste  le  chiffre  1 ; 
ce  fut  done,  le  premier  jour  de  la  semaine,  Dimanche* 

N.  B.— Les  eiecles  pairs  non-divisibles  par  le  chiffre  400  ne  sont  pa*- 
des  ann6es  bissextiles. 


CALENDA.ll. 


85 


an 


fcen  £aa  <*wf  Me 


9Jtet!jobe.  —  Streid)  bie  ©ieben  au§  Don  bie  beiben 
Ie|ten  ftummern  auf  bag  3>afjr,  bet  SKinuent  »on  ben 
beiben  le^ten  5ftummern  im  Saljre,  bit)toirt  bet  bier— 
gebraucfye  ni^t  ben  tReft—  ben  ®atum  auf  ben  SDJtonat, 
unb  bie  $tgur  auf  ba§  Sa^r.  28a§  ilberbleibt  ift  bet 
2ag  inber  ZBodje,  ber  crfte  ©onntag,  bet  jmeite 
tag  u*  f,  m. 


0  oor  3U^ 


giguren  t)or  btc  donate. 

^  »or  @cpt.  it.  ®ecf>r.    3  oor  3att.  u.  Oct.  5  wor  3luguft. 
2  oor  Stpril  itnb  Sitlu    4  oor  3tfai.         '    6  oor  §eb., 


3)er  Saturn  im  S^uar  unb  ^c&riiar  ift  ein  wentger 

SDatum  auf  bie  3afjre. 


l,t|ibt 
% 


e  unb  30te  Sal) 


igur  nor  bag  2te,  9te  unb  16te  3a&' 
Ite,  8te,  15te,  I8te,  22 
3te,  7te,  I4te  Sa^ui 
6te;  I3te,  17te,  21te,  25te,  29te  3a^ 
5te,  12te,20te,  24te,  unb  28te  3<4 
4tc  nnb  11  3ol)r^unbeii. 
Stc,  lOte,  I9te,  23te  unb  27te  ^a^r^unbert. 


©jempeL — 2BeId)er  Sag  in  ber  2Bod)e  tDar  ber  31. 
Sluguft,  1873  ?  Slntmort  ©onntag. 

ffitc  le^ten  beiben  SiQuren  im  3a^re,    73  —  70  ==  3 
5Jlinuent  auf  bo.  -f  bei  bier,  18  +  3  —  21=0 

Datum  im  2Monat,  3 1  —  28  =  3 

Stgur  auf  ben  SJlonat,  5  +  3  —  7  =  i 

£)er  9feft  1  geigt  @ud)  ben  erften  Sag  in  ber  2Bod)e, 
tt>el$er  ift  ©onntag* 


NOIE  — Between  the  Julian  and  the  Augustan  Calendars  there 
was  a  difference  of  ten  days  in  1583  and  of  eleven  days  in  1753. 
At  the  present  time  the  difference  is  twelve  days.  The  latter 
came  into  use  in  Catholic  countries  in  J683  and  in  England  ia 
1753. 


86          HOWARD'S  ART  OF  COMPUTATION. 

Howard's   California  Calendar  for  Thirty 
Centuries. 

EULE. — Cast  all  the  sevens  out  of  the  last  two 
figures  of  the  year ;  add  the  remainder  to  ihe  quotient* 
of  the  last  two  figures  of  the  year,  divided  by  four ; 
take  this  sum  with  the  day  of  the  month,  the  figure 
for  the  month,  and  the  figure  for  the  century,  dropping 
all  the  sevens  as  they  occur,  one  remainder  will  be  the 
the  first  day  of  the  week,  Sunday ;  2,  the  second,  &c. ; 

0,  last  day  of  the  week,  Saturday. 

,  *  Disregard  the  fraction,  if  any,  in  the  quotient. 

TABLE    OF    FIGURES    FOR    THE    MONTHS. 

1,  Sept.  and  Dec.      8,  Jan.  and  Oct.       5,  August.  0,  June. 

2,  April  and  July.    4,  May.  6,  Feb.,  March,  Nov. 

NOTE.— The  figure  for  January  is  2,  and  February  6  in  leap  year. 
TABLE   OF  FIGURES  FOR  THE  CEKTURIES. 

1,  IB  the  figure  for  the  2d,  Oth,  and  16th  centuries. 

2,  i4  l  1st,  8th,  15th,  18th,  23d,  26th,  30th  centuries, 

3,  "  '  7th,  I4th  centuries. 

4,  "  '  6th,  13th,  17th,  21st,  25th,  29th  centuries. 

5,  %k  *  5th,  12th,  20th,  24th,  28  centuries. 

6,  "  '  4th,  llth  centuries. 

0,  "  "    8d,  10th,  19th,  23th.  27th  centuries. 

EXAMPLE. — What  day  of  the  week  was  the  31st 
August,  1873?  Sunday,  Ans. 

Process — 

Last  two  figures  of  the  year,  73  —  70  =  3 
Quotient  of  73     -5-  by  four,    18+3  —  21=0 
Day  of  month,  31  — 28  =  3 

Figure  for  the  month,  5  +  3  —  7  =  1 


After  casting  out  the  sevens  the  remainder  is  1 : 
hence  it  was  on  the  first  day  of  the  week,  Sunday. 

iq.  B.— The  even  centuries  not  divisable  by  400  are  not  leap  years. 


SQUARE  AND  CUBE  ROOT,  87 

SQUARE  AND  CUBE  ROOT. 

1.  A  square  number  multiplied  by  a  square  num- 
ber, the  product  will  be  a  square  number. 

2.  A  square  number  divided  by  a  square  num- 
ber, the  quotient  is  a  square. 

3.  A  cube  number  multiplied   by  a   cube,  the 
product  is  a  cube. 

4.  A  cube  number  divided  by  a  cube,  the  quo- 
tient will  be  a  cube. 

5.  If  the  square  root  of  a  number  is  a  composite 
number,  the  square  itself  may  be  divided  into  inter 
ger  square  factors ;  but  if  the  root  is  a  prime  num- 
ber, the   square  cannot  be  separated  into  square 
factors  without  fractions. 

6.  If  the  unit  figure  of  a  square  number  is  5, 
we  may  multiply  by  the  square  number  4,  and  we 
shell  have  another  square,  whose  unit  period  will 
be  ciphers. 

7.  If  the  unit  figure  of  a  cube  is  5,  we  may  mul- 
tiply by  the  cube  number  8,  and  produce  another 
cube,  whose  unit  period  will  be  ciphers. 

8.  If  a  supposed  cube,  whose  unit  figure  is  5,  be 
multiplied  by  8,  and  the  product  does  not  give  3 
ciphers  on  the  right,  the  number  is  not  a  cube. 

To  prove  cube  root :  from  a  cube  number  subtract  its 
root;  the  remainder  will  be  a  multiple  of  6. 

From  a  number  that  is  not  a  cube,  subtract  the  ascer- 
tained part  of  its  cube  root;  divide  the  difference  by  6; 
then  divide  the  remainder  in  the  example  by  6 ;  the  ex- 
cess, if  any,  should  in  each  case  be  the  same. 


88          HOWARD'S  ART  OF  COMPUTATION. 

TABLE 

For  comparing  the  natural  numbers  with  the  unit 
figure  ol  their  squares  and  cubes.  By  the  use  ot 
this,  many  roots  may  be  extracted  by  observation: 

Numbers.  ..123456789  10 
Squares....  14  9  16  25  36  49  64  81  100 
Cubes 1  8  27  64  125  216  343  512  729  1000 

The  product  of  a  number  taken  any  number  of 
times  as  a  factor,  is  called  a  power  of  the  number. 

A  root  of  a  number  is  such  a  number  as  taken 
some  number  of  times  as  a  factor,  will  produce  a 
given  number. 

If  the  root  is  taken  twice  as  a  factor  to  produce 
the  number,  it  is  the  square  root;  if  three  times,  the 
•cube  root;  if  four  times,  the  fourth  root. 

By  observing  the  above  table,  it  will  be  seen 
that  the  square  of  any  one  of  the  digits  is  less  than 
100,  and  the  cube  of  any  one  of  the  digits  is  less 
than  1000;  therefore,  the  square  root  of  two  figures 
cannot  be  more  than  one  figure. 

The  square  of  any  number  equals  its  root,  plus 
the  preceding  square  and  root  of  a  consecutive  series. 
48=16.  4  +  9+3=16. 

The  units  figure  in  the. cube  root  of  a  perfect  cube 
is  the  units  figure  in  the  product  of  the  units  figure 
of  the  cube  multiplied  twice  into  itself. 

Find  the  cube  root  of  343. 

The  units  figure  3  X  3  X  3=27.     Ans.  7. 

The  difference  of  the  squares  of  two  numbers 
equals  their  sum  multiplied  by  their  difference. 


SQUARE    BOOT.  89 

To  find  the  square  root  of  a  number. 

EULE  1.  Separate  the  given  number  into  periods 
of  two  figures  each,  beginning  at  the  unit's  place. 

The  number  of  figures  in  the  root  equals  the  number  of  periods. 

2.  Find  the  greatest  number  whose  square  is  con- 
tained in  the  period  on  the  left ;  this  will  be  the 
first  figure  in  the  root.  Subtract  the  square  of  this 
figure  from  the  period  on  the  left;  to  the  remainder 
annex  the  next  period  to  form  a  dividend. 

.3.  Divide  this  dividend,  omitting  the  figure  on 
the  right,  by  double  the  part  of  the  root  already 
found,  and  annex  the  quotient  to  that  part,  and 
also  to  the  divisor ;  then  multiply  the  divisor  thus 
completed  by  the  figure  of  the  root  last  obtained, 
and  subtract  the  product  from  the  dividend. 

4.  If  there  are  more  periods  to  be  brought  down, 
continue  the  operation  in  the  same  manner  as  be- 
fore. 

NOTE  1.  If  a  cipher  occurs  in  the  root,  annex  a  cipher  to  the  trial  di- 
visor, and  another  period  to  the  dividend,  and  proceed  as  before. 

2.  If  there  is  a  remainder  after  the  root  of  the  last  period  is  found, 
annex  periods  of  ciphers,  and  continue  the  root  to  as  many  decimal 
places  as  are  required. 

.EXAMPLE. — Find  the  square  root  of  1016064. 

1,01,60,64(1008 
1 


2008)    016064 
16064 


NOTE.     The  square  root  of  a  fraction  may  be  found  by  extracting  the 
square  root  of  the  numerator  and  denominator  separately. 


90          HOWARD'S  ART  or  COMPUTATION. 
To  find  the  cube  root  of  a  number. 

RULE  1.  Beginning  at  the  units'  place,  separate  the 
given  number  into  periods  of  three  figures  each;  the 
number  of  figures  in  the  root  will  be  equal  to  the  num- 
ber of  periods. 

2.  Find  the  greatest  number  whose  cube  is  contained 
in  the  left-hand  period ;  this  will  be  the  first  figure  in  the 
root ;  subtract  its  cube,  and  to  the  remainder  annex  the 
next  period. 

3.  Multiply  the  ascertained  part  of  the  root  by  3,  then 
multiply  that  result  by  the  first  figure  in  the  root,  the 
product  with  two  ciphers  annexed  is  the  first  trial  divisor. 

4.  Find  how  many  times  the  divisor  is  found  in  the  div- 
idend and  place  the  result  in  the  root,  and  also  to  the  right 
of  the  first  terrain  the  left  hand  column;  multiply  the 
last  result  by  the  new  figure  in  the  root  and  add  the  pro^ 
duct  to  the  trial  divisor;  the  sum  is  the  complete  divisor. 

5.  Multiply  the  complete  divisor  by  the  second  figure 
in  the  root,  subtract  the  product  from  the  dividend  and 
bring  down  the  next  period. 

6.  To  find  the  next  trial  divisor  add  the  square  of  the 
last  found  figure  in  the  root  to  the  preceding  divisor  and 
its  smaller  part;  to  the  sum  annex  two  ciphers,  complete 
the  divisor  as  before. 

7.  Repeat  the  foregoing  process  with  each  period  until 
the  exact  root,  or  a  sufficient  approximation  to  it  is  found. 

EXAMPLE. — Find  the  length  of  one  edge  of  an  excava- 
tion from  which  a  cubic  mass  of  earth  =  1,745,337,664 
cubic  feet  is  to  be  taken.  Ans.  1204  feet. 


32 
1st  complete  divisor, 

3604 
2nd  com.  divisor. 

300 
64 

1,745,337,664(1204,  cube 
1                                     root. 

364 
4,320,000 
14,416 

745 

728 

17,337,664 
17,337,664 

4,334,416 

NOTE  1.— If  a  cipher  occurs  in  the  root,  annex  two  ciphers  to  the  trial 
divisor  and  another  period  to  the  dividend,  and  then  proceed  as  before. 

2.  If  there  f s  a  remainder,  after  the  root  of  the  last  period  is  found, 
annex  periods  of  ciphers  and  proceed  as  before  to  as  many  decimal  places 
as  the  answer  requires. 

8.  The  cube  root  of  a  fraction  may  be  found  by  extracting  the  cube 
root  of  the  numerator  and  denominator. 


THE  NUMBER  .NINE.  ui 

CASTING     OUT    THE    NINES. 

The  number  nine  has  many  peculiar  properties 
in  our  system  of  notation.  Any  number  is  divisi- 
ble by  9  when  the  sum  of  its  digits  is  divisible  by  9. 

Any  remainder  left  after  dividing  a  number  by  9, 
will  be  left  after  dividing  the  sum  of  its  digits  by  9. 

This  peculiarity  may  be  used  with  advantage  in 
proving  the  four  fundamental  rules,  by  casting  out 
the  nines,  that  is,  dropping  9  whenever  the  sum 
reaches  or  exceeds  that  number,  thus  to  cast  the 
9s  out  of  846732,  we  say  8+4  less  9  leaves  3;  3+6 
less  9  leaves  0 ;  7+5  less  9  leaves  3 ;  hence  the 
following. 

To  prove  ADDITION,  cast  out  the  nines  from  the 
example,  and  from  the  ascertained  sum,  if  correct 
the  excess  in  each  will  be  the  same. 

To  prove  SUBTRACTION,  the  excess  of  the  re- 
mainder should  equal  the  excess  in  the  minuend 
less  the  excess  in  the  subtrahend. 

NOTE.  If  the  excess  in  the  minuend  is  less  than,  the  excess 
in  the  subtrahend,  it  must  be  increased  by  nine. 

To  prove  MULTIPLICATION.  The  excess  of  the 
product,  must  equal  the  product  of  the  excess  of 
the  factors. 

Note.  If  the  multiplier  or  multiplicand  is  a  multiple  of 
nine,  the  product  will  have  no  excess. 

To  prove  DIVISION.  The  excess  of  the  dividend 
must  equal  the  product  of  tho  excesses  in  Quotient 
and  Divisor,  plus  the  excess  of  the  remainder. 


92          HOWARD'S  ART  OF  COMPUTATION. 

MARKING   GOODS. 

Removing  the  decimal  point  one  place  to  the  left 
on  the  cost  of  a  dozen  articles,  gives  the  cost  of 
one  article  with  20  per  cent,  added.  We  remove 
the  point  one  place  to  the  left,  because  12  tens 
make  120.  Hence,  to  find  the  selling  price,  to  gain 
the  required  percentage  of  profit,  we  have  the  fol- 
lowing general  rule : 

RULE. — Remove  the  decimal  point  one  place  to 
the  left  on  the  cost  per  dozen,  to  gain  20  per  cent. ; 
increase  or  diminish  to  find  the  percentage,  as  per 
following  table : 

TABLE  FOR  MARKING  ALL  GOODS  BOUGHT  BY  THE  DOZEN. 

To  make  20$  remove  the  point  1  place  to  left. 

"       25$     "  44          4i     Add  yV  foself. 

"       26$     u  4i  4i          44      4i     fa    " 

«       28$     «  «  44  '4      44     fa    4i 

44       30$     «  «  4i          "      44    -TV    " 

44       32$     "•  44  44          44      44     fa    44 


4:0%  "  "  "  u  "  -i-  " 

44^''  "•  u  u  4'  4t  .i  u 

50$  "  "  "  4-  4i  ^ .  " 

60$  "  "  a  "  4i  ^-  44 

80$  4<  4t  44  4<  a  |  4i 

12£$  "  "  44  44  subtract^  " 

16|$  4C  «  "  44  44  fa  " 

18f$  4t  "  44  44  44  fa  u 


REFERENCE  TABLES, 


MULTIPLICATION  TABLK. 


1 

2 

3 

4 

5 

6 

7 

8 

g 

10 

11 

12 

2 

4 

a 

6 

9 

4 

8 

12 

16 

5 

10 

15 

20 

25 

6 

12 

18 

24 

30 

36 

7 

14 

21 

28 

35 

42 

49 

8 

16 

24 

32 

40 

48 

56 

64 

9 

18 

27 

36 

45 

54 

63 

72 

81 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

ABBREVIATIONS  rssu  IN  BUSINESS. 


@ At. 

%  or  Acc't .  .Account. 

Ain't. Amount. 

Ass'd Assorted. 

Bal Balance. 

Bbl.. .; Barrel. 

B.  L Bill  of  Lading. 

%   Per  cent. 

Co —  Company. 

C.  0.  D .  . .  Collect  on  Delivery. 
Cr ....Creditor. 

Com Commission. 

Cous't Consignment. 

Cwt Hundred  Weight. 

Dft Draft. 

Disct Discount. 

Do The  same 

Dox.. Dozen. 

Dr Debtor. 

E.  E Errors  excepted. 

Ea Each. 

Exch Exchange. 

Exps Expenses. 

Fol Folio. 

Fw'd Forward. 

Fr't Freight. 


Gtiar Guarantee. 

Gal Gallon. 

Hhci     Hogshead. 

Ins Insurance. 

Inst This  month. 

Iiivt Inventory 

Int Interest. 

Mdse Merchandise. 

Mo Month. 

Net Without  disc't. 

No Number. 

Pay  *t Payment. 

Pd Paid. 

Per  An By  the  year. 

Pk'gs Packages. 

Per By. 

£,,s,,d,  Pounds,  shil'gs,  pence. 

Prem Premium. 

Prox Next  month. 

Ps Pieces. 

Rec'd Received. 

R.  R Railroad. 

Ship't Shipment. 

Sund's Sundries. 

S.  S Steamship. 

Ult Last  month. 


94 


SPECIFIC  GRAVITY. 


Specific  Gravity  is  the  weight  of  a  body  compared  with  another 
of  the  same  bulk  taken  as  a  standard.  The  exact  weight  of  a  cubic 
inch  of  gold,  compared  with  a  cubic  inch  of  water,  is  called  its  SPE- 
CIFIC GRAVITY.  Water  is  the  standard  for  solids  and  liquids.  A  cu- 
bic foot  of  rain  water  weighs  1000  ounces. 

NOTE. — To  find  the  weight,  in  ounces,  of  one  cubic  foot  of  any 
substance  here  named,  remove  tha  decimal  point  three  places  to 
the  right. 

Acid,  Acetic 1.008 

Acid,  Arsenic . . . 3.391 

Acid,  Nitric 1.271 

Air 001 

Alcohol,  of  Commerce, . . .  .885 

"         Pure, 794 

Alderwood, 800 

Ale, 1.085 

Alum, 1.724 

Aluminum, 2.560 

Amber 1.064 

Amethyst 2.750 

Ammonia, 875 

Aeh, 800 

Blood,  Human, 1.054 

Brass,  (about) 6.000 

Brick, 2.000 

Butter, 942 

Cherry, 715 

Cider, 1.018 

Coal,  bituminous,  (about)  1.250 

"     anthracite, 1.500 

Copper, 8.788 

Coral, 2.540 

Cork, 340 

Diamond 3.530 

Earth  (mean  of  the  Globe)  5.210 

Elm, 671 

Emerald, 2.878 

Ether. 632 

Fat  of  Beef, 923 

Fir, 550 

Glaeo  plate, 2.760 

Gold,  hammered, 19.362 

"    Coin, 17.647 

Granite 2.625 


1.987 

Gunpowder, 900 

Gum  Arabic 1.462 

Gypeum, 2.288 

Hazel 600 

Hematite  Ore, 4.705 

Honey, 1.456 

Ice, 930 

Iodine, 4.948 

Iridium, 28.000 


Iron, 7.645 

"     Ore, 4.900 

Ivory,  1.917 

Lard, 947 

Lead,  cast,  11.350 

"      white, 7.235 

Lignum  Vitao 1.333 

Lime 804 

*k     stone, 2.386 

Mahogany, 1.063 

Malachite 3.700 

Maple, 750 

Marble, 2.716 

Men  (Living.) 891 

Mercury,  pure, 14.000 

Mica, 2.750 

Milk 1.032 

Naptha 700 

Nickel 8.279 

Nitre 1.900 

Oak, 1.170 

Oil,  Castor 970 

Opal,.. 2.314 

Opium, 1.337 

Pearl, 2.510 

Pewter, 7.471 

Platinum  Wire, 21.041 

Poplar, 383 

Porcelain 2.385 

Quartz, 2.500 

Rosin, 1.100 

Salt 2.130 

Sand 1.750 

Silver  coin, 10.534 

Slate, 2.110 

Steel, 7.816 

Stone 2.500 

Tajlow, 941 

Tin 7.291 

Turpentine,  spirits  of, 870 

Walnut 671 

Water,  distilled, 1.000 

Wax. 897 

Willow, 585 

Wine, 992 

Zinc,  cast, 7.190 


FORMULAS. 


95 


The  Diameter 
of  a  Circle. 


1 


f 


X.8862 
—1.1284 
X.866 
-H.1547 
X.707 
-7-1.4142 
X.3183 
-7-3.1416 
X.2821 
-7-3.545 

The  Circumfer-  J    x  .2756 
ence  of  a  Circle.  S   _^3.6276 
X.2251 
-j-4.4428 
X.  15915 


X  3<if  Jo     1  =The  Circumference. 

—;-    .OiO.J          J 


\  =The  side  of  an  equal 
f         Square. 


The  Area  of  a 
Circle. 


The  Surface  of 
a  Sphere 


The  Volume 
of  a  Sphere 

The  Diameter / 

of  a  Sphere  ~~  \ 

The  Circum- 
of  a  Sphere 


-i-3.1416 
X  1.2732 

-i-  .7854 
X  12.5663 
-r-  .07958 


\  =The  side  of  an  inscribed 
J  Equilateral  Triangle. 
\  =The  side  of  an  inscribed 
I  Square. 

j  =The  Diameter. 

\  =The  side  of  an  equal 
/         Square. 

"I  =The  side  of  an  inscribed 
J  Equilateral  Triangle. 
)  =The  side  of  an  inscribed 
/  Square. 

-M5.2831S  }=The  Radius. 

=The  square  of  Radius. 
>  =The  square  of  Diameter, 

\  =The  square  of  Circum- 
/  ference. 

Circumference  X  its  Diameter. 
(Radius)2  X  12.5664 
(Diameter)2  X  3.1416 
(Circumference)2  X  .3183 
Surface  X  1-6  its  Diameter. 
(Radius)3  X  4.1888 
(Diameter)3  X  .5236 
(Circumference)3  X  .0169 
v/  of  Surface  X  .5642 
^/  of  Volume  X  1.2407 
v/  of  Surface  X  1 .77255 
&  of  Volume  X  .38978 
The  Radius  of__  j  \f  of  Surface  X  .2821 
a  Sphere        =  \  &  of  Volume  X  .6204 
The  Side  of  an^  j  Radius  X  1.1.547 
inscrib'dCube      \  Diameter  X  .5774 
TRIANGLES. — The  area=the  base  X  half  the  altitude. 
The  hypothenuse=  x/of  the  sum  of  the  squares  of  the 
base  and  the  perpendicular. 

The  base,  or  perpendicular=thev/of  the  difference  be- 
tween the  square  of  the  hypothenuse  and  the  square  of 
the  given  side. 


96 


LATITUDE  AND  LONGITUDE. 


Longitude  reckoned  from  the  Meridian  of  Greenwich. 

NORTH  AND   SOUTH  AMERICA. 


Place. 

Lat. 

Long. 

Place. 

Lat. 

Long. 

Albany,  N.  Y.. 
AnnArbXMich 
Annapolis,  Mel. 
Augusta,  Me.  .  . 
Austin,  Texas. 
Baltimore,  Md. 
Bangor,  Me  — 
Boston,  Mass.. 
Brooklyn,  N.  Y. 
Buffalo,  N.  Y.. 
Burlington,  Vt. 
Buenos  Ayres. 
Cambr'ge,  Mass 
Cape  May,  N.  J. 
Cape  Horn  
Charleston,  S.C 
Chicago,  111  
Cincinnati,  O.  .. 
Columbia,  S.  C. 
Concord,  N.  H. 
Council  Bluffs. 
DesMoines,  lo. 
Detroit,  Mich.  . 
Dover,  Del  
Dubuque,  lo.  .  . 
Fred'csb'rg,  Va 
Fort  Laramie.  . 
Ft.  L'v'wth,  Ks. 
Frankfort,  Ky.. 
Galveston,  Tex 
Georgetown, 
Bermuda,  W.I. 
Guayaquil  
Havana 

o    / 

42  40  N 
42  17 
38  59 
44  19 
30  13 
39  18 
44  48 
42  21 
40  42 
42  50 
44  27 
34  36s 
42  23  N 
38  56 
55  59  s 
32  47  N 
41  54 
3906 
34 
43  12 
41  30 
41  35 
4220 
39  10 
4230 
38  18 
42  12 
3921 
38  14 
29  18 

32  22 
2  13  s 
23    9N 
44  39 
40  16 
41  46 
39  55 
38  36  N 
24  33 

0      / 

73  45w 
83  43 
76  29 
69  50 
97  39 
76  37 
68  46 
71  03 
73  58 
7859 
73  10 
5822 
71  08 
7457 
07  16 
79  56 
87  38 
8430 
81  02 
71  29 
93  48 
9340 
83    2 
75  30 
9040 
77  27 
104  48 
94  44 
84  40 
94  47 

64  37 
79  53 
82  21 
63  35 
76  50 
72  41 
86    5 
92    8 
81  47 

Lima,  

o    / 

12    3  8 

34  40  N 
38    3 
19  26 
43    2 
30  41 
45  31 
41  18 
29  58 
40  43 
45  23 
39  57 
37  14 
43  39 
41  49 
46  40 
37  32 
43    8 
22  56  s 
32    5N 
38  35 
29  48 
38  37 
44  53 
40  46 
37  48 
35  41 
3948 
39  40 
43    3 
43  31 
40  13 
42  44 
33    2s 
38  53  N 
41  23 
40    7 
34  14 
42  16 
37  13 

0      / 

77    6 
92  12 
85  30 
99    5 
8754 
88    1 
73  33 
72  55 
90    2 
74 
75  42 
75    9 
7724 
70  15 
71  24 
71  12 
7726 
77  51 
43    9 
81    5 
121  28 
81    5 
90  15 
95    5 
1126 
122  47 
106  1 
89  33 
94  52 
76    9 
7923 
74  45 
73  41 
71  41 
77    0 
73  57 
8042 
77  57 
71  48 
76  34 

Little  Rock,  Ark... 
Louisville,  Ky  
Mexico,  Mexico  — 
Milwaukee,  Wis.  .  . 
Mobile  Ala 

Montreal,  C.  E  
New  Haven,  Conn.. 
New  Orleans,  La... 
New  York,  N.  Y... 
Ottawa  C  W 

Philadelphia,  Pa... 
Petersburg,  Va  — 
Portland  Me  

Providence,  R.  I... 
Quebec.  C.  E  
Richmond,  Va  
Rochester,  N.  Y... 
Rio  Janeiro 

Savannah,  Ga  
Sacramento,  Cal.  .  . 
St.  August'e,  Fla.. 
St  Loui**  Mo 

St.  Paul.  Minn  
Salt  Lake  City  
San  Francisco  
Santa  Fe,  N.  Mex.. 
Springfield,  111  
St.  Joseph's,  Mo... 
Syracuse,  N.  Y  — 
Toronto,  C.  W  
Trenton,  N.  J  
Troy  N  Y 

Valparaiso,  
WASHINGTON  
West  Point,  N.  Y.. 
Wheeling,  W.  Va.. 
Wilmington.  Del... 
Worcester,  Mass... 
Yorktown,  Va  

Halifax 

Harrisburg,  Pa. 
Hartford,  Conn 
Ind'nap'lis,  Ind 
Jeffer1  City,  Mo 
Key  West,  Fla, 

A  difference  of  15  degrees  of  Longitude  equals  a,  difference  of  one  hour  of  time. 

The  degrees  of  Longitude  between  two  cities,  multiplied  by  4, 

equals ,  in  minutes,  the  difference  of  time. 

For  »  difference  of  There  \a  ft  difference  of  For  a  difference  A          There  is  »  difference  of 

15°  in  Long.       1  hr.  in  Time.  1°    "      "  4min."      " 

15'    "      "  1  min."      "  1'     "      "  4  sec.  "      u 

15"  "      "  1  sec.  "      "  V    "      "  1-15  sec.  in  tim* 


LATITUDE  AND  LONGITUDE. 


97 


EUROPE,  ASIA,  AFRICA,  AND  THE  OCEANS. 


Place. 

Lat. 

Long. 

Place. 

Lat. 

Long. 

Antwerp  
-Alexandria 

o     / 

51  13  N 
31  12 
6432 
37  58 
36  11 
36  47 
52  22 
5 
34    2 
41  23 
18  56 
53    5 
52  30 
50  51 
51  26 
50  58 
41    1 
23    7 
59  59 
55  41 
33  56  s 
2234N 
37  54 
30    3 
9  49 
53  23 
51    8 
55  57 
17  41  s 
43  46  N 
51  29 

0       / 

424E 
29  53 
40  33 
23  44 
37  10 
3    4 
4  53 
115 
151  13 
2  11 
72  54 
8  49 
13  24 
4  22 
9  29w 

1  51  E 

28  59 
113  14 
29  47 
12  34 
18  29 
88  20  E 
22  52 
31  18 
80  23 
6  20w 

1   19  E 

3  12w 
17853E 
11  16 

Leghorn 

0       / 

4332 
51  20 
3842 
55  40 
35  54 
38  12 
13  20 
23  37 
43  18 
14  36 
14    4 
40  25 
36  43 
84  24  s 
1528 
3436N 
4050 
46  28 
39  54 
38    8 
48  50 
41  54 
51  54 
38  26 
1  17 
14  55 
8  30 
15  55  s 
29  59  N 
59  21 
59  56 
43  07 
34  54 
36  47 
35  47 
4550 
48  13 
52  13 
6  28s 

0       / 

10  18K 
1222 
9    9\v 
35  33  E 
14  30 
15  35 
43  12 
58  35 
5  22 
121  2 
80  16 
3  42w 
4  26 
173  IE 
1677 
138  51 
14  16 
30  44 
116  28 
13  22 
2  20 
12  27 
4  29 
27    7 
103  50 
100 
13  18w 
5  45 
32  34  K 
18    6 
30  19 
5  22 
13  11 
10    6 
554 
12  26 
1623 
21    2 
3933 

Leipsic       

Archangel 

Lisbon 

Athens 

Moscow  

Aleppo 

Malta  

Algiers    

Messina  .  *  

Amsterdam  

Mocha  
Muscat  

Borneo 

Botany  Bay 

Marseilles     .   .  . 

Barcelona  
Bombay 

Manilla  

Madras 

Bremen 

Madrid  

Berlin. 

Malaga  

Brussels  

New  Zealand  
New  Hebrides..  . 

Cape  Clear 

Calais  

Constantinople  .  . 
Canton  

Naples 

Odessa  

Cronstadt 

Pekin 

Copenhagen  
Cape  of  G.  Hope. 
Calcutta 

Palermo  
Paris  

Rome 

Corinth  

Rotterdam.. 

Cairo 

Smyrna 

Ceylon 

Singapore 

Dublin  

Siam  

Dover  

Sierra  Leone  
St.  Helena 

Edinburgh  

Feejee  Group..  .  . 
^Florence 

Suez     

Stockholm 

GREENWICH  .   .  . 

St.  Petersburgh  . 
Toulon    

Geneva  

46  12 
55  52 
36    7 
44  24 
21  19 
53  33 
49  29 
31  48  N 
53  25 

6    9 
4  16w 
5  22 
8  53  E 
157  52w 
9  58  E 
6 
37  20w 
3 

Glasgow  
Gibraltar  

Tripoli  
Tunis 

Genoa  

Tangier 

Honolulu  

Venice  

Hamburg 

Vienna 

Havre  

Warsaw  

Jerusalem  . 

Zanzibar 

Liverpool  

MEASURE  OF  CIRCLES,  OR  ANGLES. 

The  UNIT  is  the  degree,  which  is   1-360  part  of  the  circumference 
of  any  circle. 

60  Seconds  (")          =  1  Minute.  ' 
60  Minutes  =  1  Degree.  ° 

30  Degrees  =  1  Sign.       S 

12  Signs,  or  360°       =  1  Circle.    C 


98               LEGAL  RATES  OF  INTEREST 

And  Statute  Limitations  in  the  different  States. 
In  some  States  there  are  exceptions,  and  any  of  the  data  are  liable  to 
change  by  the  action  of  the  State  Legislatures. 
The  English  legal  rate  is  5  per  cent. 

States  and  Territo- 
ries. 

o 

P 

•g   . 

^0 

=:  2 

1! 

OJ     K^ 
gj^ 

Penalties  for  Usury. 
Forfeiture  of 

Statute  Limitat'n. 

Open 

Acc'ts 
Yrs. 

Note. 
Yrs. 

Judg- 
ment, 
Yrs. 

Alabama,  

8 

10 

G 
10 
10 
G 
7-10 
6 
G 
8 
7 
10 
G 
6 

6 
7 
6 
5 
G 
6 
6 
7 
7 
6 
6 
10 
10 
10 
6 
7 
6 
7 
6 
6 
10 
G 
6 
7 
6 
8 
10 
6 
G 
10 
G 
7 
12 

8 

Any. 
Any. 
Any. 
Any. 

Any. 

10 
Any. 
10 
Any. 
10 
10 

10 
12 
10 
8 
Anv. 
G 
Anv. 
10 
12 
10 
10 
Any. 
12 
Anv. 
G 
7 

8 
8 
12 
Any. 
Any. 
Any. 
10 
Any. 
Any. 
6 
12 
Any. 
G 
10' 

Entire  interest 

Entire  interest 

Principal 
Entire  interest 

Excess 

Entire  interest 
Excess 

Entire  interest 

Entire  interest 
Excess 
Excess 

Entire  interest 
Entire  interest 

Thrice  excess 
Entire  interest 

Excess 
Entire  interest 
Excess 

Excess 
Excess 

Excess 
Entire  interest 

3 

3 
2 
2 

6 
G 
3 
3 
5 

3 

5 
G 

5 
3 
2 
3 
6 
3 
G 
G 
G 
3 
5 

4 

G 
6 

G 
3 
6 
G 
6 
6 
G 
6 
2 

6 

5 

5 

10 

G 

4 
4 
6 
15 
6 
3 
5 
3 

G 
20 

10 
5 
7 
5 
G 
3 
G 
6 
G 
6 
10 

5 

6 
16 

6 
3 
15 
6 
6 
G 
G 
G 
4 

G 
5 

5 

6 

20 

10 
10 
5 

ir 

6> 
20 

12 

10 
20 

20 
10 
14 
10 
20 
12 
20 
20 
10 
20 
20 

5 

20 
20 

20 
10 
20 
10 
20 
20 
20 
10 
10 

10 

10 
10 

Alaska      

Arizona,  

Arkansas,  

California 

Colorado  

Connecticut,  

Dakota  

Delaware  

Dist.  of  Columbia. 
Florida 

Georgia,  

Idaho 

Illinois    

Indiana,  
Indian  Territory,.. 
Iowa,  

Kansas,  

Kentucky,  

Louisiana,  

Maine,  

Maryland,  

Massach  usetts,  
Michigan  
Minnesota  

Mississippi,  

Missouri,  

Montana 

Nebraska,  

Nevada 

New  Hampshire,.. 
New  Jersey,  

New  Mexico,  

New  York,  

North  Carolina,.... 
Ohio 

Oregon  
Pennsylvania,  
Rhode  Island  
South  Carolina,  
Tennessee,  
Texas  

Utah 

Vermont    

Virginia,  

Washington  
West  Virginia,  .... 
Wisconsin,    

Wyoming,  

PAPER  TABLE.  99 

Paper  is  bought  at  wholesale  by  the  bale,  bundle  and  ream;  and  at  re- 
tail by  the  ream,  quire  and  sheet. 

24  Sheets  —  1  Quire,  2  "Reams    •=  1  Bundle, 

20  Quires  =»  1  Ream,  5  Bundles  —  1  Bale. 

The  names  generally  define  the  sizes.    "Writing  and  Drawing  Papers 
differ  in  size  from  Printing  Papers  of  the  same  name. 
English  sizes  differ  from  American. 

SIZE  OF  FOLDED  TAPERS,  IN  INCHES. 

BilletNote 6x8  Letter, 10x16 

Octavo  Note, 7x9  Commercial  Letter, 11x17 

Commercial  Note, 8x10  Packet  Post, 11^x18 

Packet  Note <  9x11  Extra  Packet  Post......  11^x18^ 

Bath  Note, 8^x14  Foolscap, , 12>£xl6 

FLAT  CAP  PAPERS. 

Juaw  Blank, 13x16  Medium, 18x23 

Flat  Cap, 14x17  Royal, 19x24 

Crown, 15x19  Super  Royal, 20x28 

Demy, 16x21  Imperial, 22x30 

Folio  Post, 17x22  Elephant, 22%x27% 

Check  Folio, 17x24  Columbia, 23x33,^ 

DoubleCap 17x23  Atlas, 26x33 

Extra  Size  Folio, 19x23  Double  Elephant 26x40 

SIZE  OF  PRINTING  PAPERS. 

Medium, 19x24  Double  Medium, 24x38 

Royal, 20x25  Double  Royal, 26x40 

Super  Royal 22x28  Double  Super  Royal, 28x42 

Imperial, 22x32  "          "          ««      2i)x43 

Medium-and-half, 24x30  Broad  Twelves, 23x41 

Small  Double  Medium,. . . .  24x36  Double  Imperial 32x46 

BOOKS. 

The  terms  folio,  quarto,  octavo,  duodecimo,  etc,,  indicate  the  number 
of  leaves  into  which  a  sheet  of  paper  is  folded. 

y  hen  a  sheet   •>    The  Book    $       1  sheet  of       When  a  sheet  )  The  Book  $     1 .  sheet  of 
js  folded  into   $     is  called    j  Paper  makes       is  folded  into  )  is  called  i  Paper  makes 

2  leaves.    A  Folio.  4  pages  16  leaves.  A  16mo.  32  pages* 

4     "          A  Quarto  or  4to.       8     "  18       "  AiilSmo.  36    " 

8     "          An  Octavo  or  8vo.  16    "  24       "  A  24mo.  43    " 

12     "  A  Duodecimo  or  12mo. 24  "••  32       "  A  32mo.  64    •• 

Clerks  and  Copyists  are  of  ten  paid  by  the  Folio  lor  making  copies  of 
legal  papers,  records  and  documents. 

72  words  make  1  folio  or  sheet  of  Common  Law. 

90     "          "      *     "      "       •*     "    Chancery. 

A  Folio  varies  in  different  States  and  Countries  but  usually  contains 
from  75  to  100  words. 


100 


GOLD  COINS. 


GOLD  COINS — their  weight,  fineness,  and  value  in. 
British  and  United  States  money,  based  on  U.  S. 
Mint  assays,  1879,  computed  by  C.  FRUSHER 
HOWARD. 


Country. 

Denomination. 

We 
Grains. 

ight. 
Ouncei. 

Fine 
leOOths 

ness. 
Carats. 

Value 
£    s.    d. 

>. 
U.S. 

$ 

Austria, 

Union  Crown, 

171.36 

0.357 

900. 

21.60 

1»  7,,  3^ 

6.6410 

Belgium, 

25  Francs, 

121.92 

0.254 

899. 

21.57 

19,,  4^ 

4.7205 

Bolivia, 

Doubloon, 

41G.16 

0.867 

870. 

20.88 

3,,  4,,  1 

15.5925 

Brazil, 

20  Milries, 

276.00 

0.575 

917.5 

22.02 

2,,  4,,  10 

10.905T 

Chili, 

Doubloon, 

416.16 

0.867 

870. 

20.88 

3,,  4,,  1 

15.5925- 

Denmark, 

10  Thaler, 

214.96 

0.427 

895. 

21.48 

M2,,  $y2 

7.9000 

England, 

Sovereign, 

123.21 

0.2567 

916.6 

22.00 

i,,  o,,  o 

4.SGG5- 

France, 

20  Francs, 

99.60 

0.2075 

899. 

21.57 

15,,10^ 

3.8562 

Germany, 

20  Marks, 

122.90 

0.256 

900. 

21.60 

19,,  6^ 

4.7627 

Greece, 

20  Drachms, 

88.80 

0.185 

900. 

21.60 

14,,  1% 

3.4419 

India, 

Mohur, 

179.52 

0.374 

916. 

22.00 

1»  9,,  1 

7.0818 

Italy, 

20  Lire, 

99.36 

0.207 

898. 

21.55 

15,,  9*4 

3.8420 

Japan, 

5  Yen, 

128.30 

0.267 

900. 

21.60 

1,,  0,,  5 

4.9674 

Mexico, 

Doubloon, 

416.16 

0.8675 

870.5 

20.89 

3,,  4,,  Ifc 

15.6105 

" 

20  Pesos, 

518.88 

1.081 

873. 

20.95 

4,,  0,,  2 

19.5083. 

Netherl'ds. 

10  Guilders, 

103.72 

0.216 

899. 

21.57 

16,,  5 

3.995S 

Peru, 

Doubloon, 

416.16 

0.867 

868. 

20.83 

3,,  3,,11% 

15.556T 

• 

20  Soles, 

496.80 

1.035 

898. 

21.55 

3,,18,,11^ 

19.2130 

Portugal, 

Gold  Crown, 

147.84 

0.308 

912. 

21.88 

1,    3,,10H 

5.806S 

Rome, 

2J  Scudi, 

67.20 

0.140 

900. 

21.60 

10,,  8 

2.6047 

Russia, 

5  Roubles, 

100.80 

0.210 

916. 

22.00 

16,,  4 

3.9764 

Spain, 

100  Reales, 

128.64 

0.268 

896. 

21.50 

1»  0»  5 

4.9639> 

Sweden, 

Ducat, 

53.28 

0.111 

975. 

23.40 

9,,  2 

2.2372 

Turkey, 

100  Piasters, 

110.88 

0.231 

915. 

21.96 

17,,UH 

4.3695 

United    ( 

20  Dollars, 

516.00 

1.075 

900. 

21.60 

4,,  2,,  zy2 

20.0000 

States.  ) 

One  Dollar. 

25.80 

.05375 

900. 

21.60 

.205486 

1.0000 

The  Gold  Talent  of  Scripture— £  5  8  64 
"    Silver   "  "          —£341 

Exactly  the  existing  ratio  between  U. 


5,,8  — $26592.809. 
,,10  ,4  —  $  1662.025. 
S.  Gold  and  Silver  Coins-16  to  1. 


SILVER  COINS.  101 

Table  of  various  Silver  Coins,  showing  their  weight, 
fineness  and  quota  of  pure  silver,  computed  from 
U.  S.  Mint  assays,  by  C.  FRUSHER  HOWARD. 


Country. 

Denomination. 

Pine- 
ness. 

Ve 
Ounces. 

ght. 
Grains. 

Pure! 
Grains. 

ilver. 
Ounces. 

Austria, 

New  Florin, 

.900 

0.397 

190.56 

171.504 

.357300 

« 

"    Dollar, 

.900 

0.596 

286.08 

257.472 

.536400 

Belgium, 

5  Francs, 

.897 

0.803 

385.44 

345.739 

.720291 

Bolivia, 

New  Dollar, 

.9035 

0.643 

308.64 

278.856 

.580950 

Brazil, 

Double  Milries, 

.9185 

0.820 

393.60 

361.521 

.753170 

Canada, 

20  Cents, 

.925 

0.150 

72.00 

66.666 

.138750 

Cen.  America. 

Dollar, 

.850 

0.866 

415.68 

353.328 

.736100 

Chili, 

New  Dollar, 

.9005 

0.801 

384.48 

346.224 

.721300 

China,  Hong  K. 

English  Dollar, 

.901 

0.866 

415.68 

374.527 

.780266 

Denmark, 

Two  Rigsdaler, 

.877 

0.927 

444.96 

390.230 

.812979 

England, 

New  Shilling, 

.9245 

0.1825 

87.60 

80.986 

.168721 

France, 

5  Franc, 

.900 

0.800 

384.00 

345.6 

.720000 

Germany, 

Mark, 

.900 

0.1785 

85.70 

77.13 

.160650 

Greece, 

5  Drachms, 

.900 

0.719 

345.12 

310.608 

.647100 

East  Indies, 

Rupee, 

.916 

.0.374 

179.52 

164.44 

.342584 

Japan, 

New  Dollar, 

.900 

0.875 

420.00 

378.000 

.787500 

Mexico, 

<«      « 

.903 

0.8675 

416.40 

376.009 

.783352 

Naples, 

Seudo, 

.830 

0.844 

405.12 

336.249 

.700520 

Holland, 

2}  Guildeis, 

.944 

0.804 

385.92 

364.308 

.758976 

-  Norway, 

Specie  Daler, 

.877 

0.927 

444.96 

390.229 

.812979 

Peru, 

Dollar  1858, 

.909 

0.766 

367.68 

334.221 

.696294 

Rome, 

Scudo, 

.900 

0.864 

414.72 

373.248 

.777600 

Russia, 

Rouble, 

.875 

0.667 

320.16 

280.140 

.583625 

Spain, 

New  Pistareen, 

.899 

0.166 

79.68 

71.632 

.149234 

Sweden, 

Rix  Daler, 

.750 

1.092 

524.16 

393.120 

.819000 

Turkey, 

20  Piasters, 

.830 

0.770 

369.60 

30G.765 

.639100 

Tuscany, 

Florin, 

.925 

0.220 

105.60 

97.680 

.203500 

United  States. 

Dollar 

.900 

0.8594 

412.50 

371.25 

.7734375 

i<        « 

Trade      " 

.900 

0.875 

420.00 

378.00 

.787500 

102 


GOLD  VALUE  OF  SILVER. 


Table  showing  the  value  in  U.  S.  Gold  Coin  of  an 
ounce  of  silver,  (480  gr.)  a  trade  dollar  (420  gr.), 
and  a  Standard  dollar  (412J  gr.),  all  9-10  fine, 
at  London  quotations  for  Silver  bullion  .9245  fine, 
calculated  at  the  par  of  exchange,  $4.8665,  to 
the  pound  sterling,  by  C.  FRUSHER  HOWARD. 


London 
Pence. 

Quotation 
£Ster'g 

Value  of 
Ounce. 
480  Gr'ns 

Value  of 
Trade  $ 
420  Gr- 

Value  of 
Stand  $ 
412%  G 

$ 
0.848 

London 
Pence. 

Quotation 
£  Ster'g 

Value  of 
Ounce. 
480Gr's. 

Value  of 
Trade  $ 
420  Gr 

Value  of 
Stand.  $ 
412^  G 

50 

.2083 

* 
0.9869 

0.864 

55% 

.2302 

$ 
1.0905 

$ 

.954 

% 
.937 

50% 

.2094 

0.9918 

0.868 

0.852 

55^ 

.2312 

1.0955 

.958 

.941 

50K 

.2104 

0.9968 

0.872 

0.856 

55% 

.2323 

1.1005 

.963 

.946 

50% 

.2115 

1.0018 

0.877 

0.861 

56 

.2333 

1.1055 

.967 

.950 

51 

.2125 

1.0067 

0.881 

0.865 

56% 

.2343 

1.1104 

.971 

.954 

51% 

.2135 

1.0117 

0.885 

0.869 

56% 

.2354 

1.1153 

.976 

.959 

WA 

.2146 

1.0166 

0.889 

0.874 

56% 

.2365 

1.1202 

.980 

.963 

51% 

.2156 

1.0215 

0.893 

0.878 

57 

.2375 

1.1252 

.985 

.968 

52 

.2167 

1.0264 

0.898 

0.882 

57% 

.2385 

1.1301 

.989 

.972 

52% 

.2177 

1.0314 

0.902 

0.887 

57^ 

.2396 

1.1350 

.993 

.976 

52% 

.2187 

1.0362 

0.907 

0.891 

57% 

.2406 

1.1399 

.997 

.980 

62% 

.2198 

1.0412 

0.911 

0.895 

58 

.2417 

1.1449 

1.002 

.985 

53 

.2208 

1.0461 

0.915 

0.899 

58% 

.2427 

1.1498 

1.006 

.989 

53% 

.2219 

1.0511 

0.919 

0.904 

58% 

.2437 

1.1548 

1.010 

.993 

53K 

.2229 

1.0560 

0.924 

0.908 

58% 

.2448 

1.1597 

1.015 

.997 

53% 

.2239 

1.0610 

0.928 

0.912 

59 

.2458 

1.1646 

1.019 

1.001 

54 

.2250 

1.0659 

0.932 

0.916 

59% 

.2469 

1.1696 

1.023 

1.005 

54% 

.2260 

1.0709 

0-937 

0.921 

59^ 

.2479 

1.1745 

1.028 

1.010 

54% 

.2271 

1.0758 

0.941 

0.925 

59% 

.2489 

1.1794 

1.032 

1.014 

54% 

.2281 

1.0807 

0.945 

0.929 

60 

.2500 

1.1844 

1.036 

1.018 

55 

.2292 

1.0857 

0949 

2.933 

London  price  per  ounce,  multiplied  by  4.S66-'>,  multiplied  by  .9,  divided 
by  .&245,  equals  the  price  per  ounce,  in  United  states  Gold  coin. 

The  Trade  dollar  is  worth  two-tenths  ol  a  cent  more  than  the  Mexican. 


STANDARD  WEIGHTS  AND  MEASURES. 


103 


HOWARD'S 
Tables  of  Standard  Weights  and  Measures. 

A  Standard  Measure  is  a  fixed  ur.it  established  by  law,  by  which 
quantity,  as  extent,  dimension,  capacity  or  value  is  measured. 

The  U.  S.  Standard  units  arc  the  YARD,  the  GALLON,  the  BLSHEL, 
the  TROY  POUND,  and  the  GOLD  DOLLAR. 

The  Standard  unit  of  weight  must  be  of  definite  dimensions,  and 
of  definite  gravity,  of  some  substance,  a  certain  volume  of  which, 
under  certain  conditions,  will  always  have  a  certain  weight. 

One  cubic  inch  of  pure  water  weighed  in  vacuo,  thermometer  62° 
Fahrenheit,  Barometer  30°=  252.458  grains. 

5760  grains  =  1  Troy  pound. 

In  the  Treasury  at  Washington  is  a  brass  scale  \\hich,  at  a  tem- 
perature of  62'J  Fahrenheit,  is  82  inches  long;  all  our  weights  and 
measures  are  referred  to  this  unit. 


LONG  MEASURE. 


SURVEYORS' 


II       LONG  MEASURE. 

IN. 

FT. 

YD. 

UD 

FUR 

12 

I 

1 

Foot 

3 

.,..1 

..1 

Yard. 

7.92 

..1 

1 

Link. 

36 

198 

16l/£ 

5'/2 

.1 

..1 

Rod. 

198 

25 

..1 

1 

Rod. 

7920 

660 

220 

40 

.    1 

..1 

Furl'ng 

792 

100 

..4 

.1 

1 

Ch'n. 

63360 

5280 

1760 

320 

'.'.S 

..1 

Mile. 

63360 

8000 

320 

80 

] 

Mile. 

The  Geographical  Mile  equals  1.15  Statute  Miles, 
COMPARISON  OF  STANDARD  MEASURES  OF  DISTANCES. 


Country. 

U.  S.  Mile. 

Country. 

U.S  Mile. 

Austria,  1  Mile, 

=  4.<)8 

Persia,  

.1  Farsang, 

=  4.17 

China  1  Li, 

=    .35 

Portugal,.... 

.IMilha, 

=  1.28 

East  Indies,  1  Coss, 

=  1.14 

Prussia,  

.IMeile, 

=  4.93 

Egypt  1  Mili, 

=  1.15 

Russia,  

.1  Verst. 

=    .66 

England, 

...IMile, 

=  1.00 

Spain  

.  1  League, 

=  4.15 

France,   . 

..  .1  Kiloinet 

r.=  .62 

Sweden,  .... 

.1MU, 

=  6.64 

Japan,  .. 

...1  Hi. 

=2.562 

Switzerland, 

1  Lieue, 

=  2.98 

Mexico,  . 

...1  Silio, 

—  6.76 

Turkey,  .... 

.IBern, 

=  1.04 

104 


SQUARE  MEASURE. 


For  measuring  land,  Boards,  Painting,  Paving,  Plastering,  etc. 


SQ.  INCH. 

SQ.  FOOT. 

SQ.  YARD, 

SQ.  RD. 

SQ.  11. 

SQ.  A. 

144 

1 

1 

SQ     FT 

1296 

9 

1 

1 

YAHD 

39204 

272*4 

30J4 

1 

1 

KOI) 

15681GO 

10890 

1210 

40 

.  .    1 

1 

HOOD 

6272640 

43560 

4840 

160 

4 

...1 

....1 

ACRE. 

4014489GOO 

27878400 

3097600 

102400 

2560 

640 

...1 

311  LE. 

lu  measuring  Roofing,  Paving,  etc.,  100  square  feet -=  one  square. 

One  thousand  shingles,  averaging 4  inches  wide,  and  laid  5  inches 
to  the  weather,  are  estimated  to  be  a  square. 

One  mile  squarc=l  section— 640  acres.  36  square  miles  (6  miles 
square)  =>!  township. 

The  sections  are  all  numbered  1  to  36,  commencing  at  the  north- 
east corner,  thus: 

The  sections  arc  all  divided  into 
quarters,  which  arc  named  by  the 
cardinal  points,  as  in  section  1. 
The  quarters  are  divided  in  the 
same  way.  The  description  of  a 
forty-acre  lot  would  read:  The 
south  half  of  the  west  half  of  the 
south-west  quarter  of  section  1  in 
township  24,  north  of  range  7  west, 
or  as  the  case  might  be ;  and  some- 
times will  fall  short,  and  some- 
times overrun  the  number  of  acres 
it  is  supposed  to  contain. 


6 

5 

4 

a 

9 

NW  |  NK 

SW  |  SK 

7 

8 

9 

10 

11 

12 

18 

17 

16* 

15 

14 

13 

19 

20 

21 

22 

23 

24 

80 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

COMPARISON  OF  THE  CoMMONT  AND  METRIC  SYSTEMS. 


1  Inch,    =        2.54  Centimeters  1  Cu.  in.= 

1  Foot.    =      30.48  Centimeters  1    "  ft..= 

1  Yard,  =  9144  Meters  1    ik  yd., 

1  Rod,     =  5.029  Meters  1  Cord, 

1  Mile,    =       1.6093  Kilometers  1  Fl.  oun 

1  Sq. in.  =6.4528  Sq.  CentimTrs  1  Gallon, 

1  Sq.  ft..  =  929  Sq.  Centimeters  1  Bushel, 

1  »•     yard,  =  8.361  Sq.  Meters  1  Troy  gr, 

1  ik     rod      =       25.29  Centairs  1      "    ID. 

1  Acre.         =  40.47  Ares.  1  Av.  Ib. 

1  Sq.  mile,   =        269  Hectares  1  Ton, 


16.39  Cn.  CentimVrs 

28320  *k 

.7646"    Meters. 

=  3.625  Stores 
ce.  =  .2.958  Centiliters 
=  3.786  Liters 
=  .3524  Hectoliter 
=  64.8  Milligrams 
=  .373  Kilo 

=  .4536     " 

=       .907  Tonneau 


CUBIC  MEASURE, 


105 


For  measuring  timber,  stone,  boxes,  packages,  capacity  of  rooms,  etc. 


CU.  IN. 

CU.  FT. 

CU.  YD. 

CD.  FT 

CD. 

PCH. 

1728 

1 

....1 

Cubic  Foot        '- 

46656 

27 

1 

....1 

Cubic  Yard 

27648 

16 

16-27 

....1 

1 

Cord  Foot 

221184 

128 

420-27 

....8 

...1 

....1 

Cord  of  "Wood 

42768 

24% 

....1 

...1 

Perch  of  Stone 

69120 

40 

....1 

TJ  S  Ton  Ship  Cargo 

One  ton  of  square  timber  =  50  cubic  feet. 

The  English  shipping  ton  =  42  cu.  ft.    The  Register  ton  =  100  cu.  ft. 
A  cord  of  wood  is  a  pile  4  ft.  high,  4  ft.  wide,  and  8  ft.  long. 
A  cord  foot  is  one  foot  in  length  of  such  a  pile. 
A  cubic  yard  of  common  earth  is  called  a  load. 
In  Board  measure  all  boards  are  assumed  to  be  1  inch  thick. 
A  board  foot  is  1  ft.  long,  1  ft.  wide  and  1  in.  thick,  hence  12 
board  feet  make  1  cubic  foot. 

Board  feet  are  changed  to  cubic  feet  by  dividing  by  12. 
Cubic  feet  are  changed  to  Board  feet  by  multiplying  by  12. 
Masonry  is  estimated  by  the  CUBIC  FOOT  and  PERCH  ;  also  by  the 

SQUARE  FOOT  and  SQUARE  YARD. 

Five  courses  of  bricks  in  the  height  of  a  wall  are  called  a  foot, 
In  board  and  lumber  measure,  estimates  are  made  on  1  inch  in 
thickness;  one-fourth  the  price  is  added  for  every  &  inch  in  thick- 
ness over  one  inch. 

MISCELLANEOUS  WEIGHTS  AND  MEASURES. 


12  Units, 1  Dozen. 

12  Dozen, 1  Gross. 

12  Gross, 1  Great  Gross. 

20  Things, 1  Score. 

196  Ibs 1  Barrel  of  Flour. 

200  "  ..1  Bbl.  Beef,  Pork,  Fish. 

66   "    1  Firkin  of  Butter. 

14    " 1  Stone,  Avoir. 

28    " 1  Quarter,    " 

2iyt  Stones, 1  Pig  of  Iron. 


SPigs, IFother. 

2  Weys  (328  Ib)  1  Sack  of  WooL 
12  Sacks,  (4368  Ib.) 1  Last 

3  Inches, 1  Palm. 

4  "      1  Hand. 

9       "      1  Span, 

3  ft 1  common  pace. 

6  "  1  Fathom. 

3  Miles, 1  League. 

360  Degrees, 1  Circle. 


106   TROY  WEIGHT. 

For  Gold,  Silver,  Jewels,  etc. 


AVOIRDUPOIS  WEIGHT. 

For  Groceries,  Provisions,  etc. 


Gr. 

Pwt. 

Oz. 

Gr. 

Oz. 

Lb. 

24 

...1 

..1 

Pennyweight 

437  1/2 

1 

..1 

Ounce 

480 

20 

..1 

..1 

Ounce. 

7000 

16 

...A 

..1 

Pound. 

5760 

240 

12 

..1 

Pound. 

14000000 

32000 

2000 

..1 

Ton. 

The  Standard  unit  is  the  Troy  Pound. 
The  Long  Ton  =  2240  Ibs.    1  cwt.  =  1121bs. 
To  compare  Troy  weights  with  Avoirdupote,  reduce  both  to  grains. 
Pounds  Avoirdupois  X  100x7-j-48=  ounces  Troy. 
Troy    ounces   X. 06  6-7=   Pounds   avoirdupois;     thafis,  ounces 
multiplied  by  .06+1-7  of  the  product. 

APOTHECARIES'  WEIGHT.        APOTHECARIES'  MEASURE. 


GRS. 

sc. 
3 

DR. 

3 

OZ. 

z 
3 

GO  Minims     =  1  Fluid  Drachm. 

8  Fl.  Drms  =  1  Fluid  Ounce. 

20 

1 

.1 

SCRUPLE. 

16  Fl.  Ozs.   —  1  Pint.. 

GO 

3 

1 

1 

DIIAM. 

8  Pints        =  1  Gallon. 

480 

v»4 

8 

1 

1 

OUNCE. 

Used  in  compounding    liquid. 

5760 

288 

% 

12 

..1 

POUND. 

medicines. 

The  grain,  ounce  and  pound  arc  the  same  as  Troy  Weight. 
Drugs  are  bought  and  sold  in  quantities  by  Avoirdupois  Wei 
1  Teaspoon  =  45  Drops.  1  Tablespoon  =  l/2  Fluid  Ounce. 

COMPARISON  OF  LIQUID  MEASURES. 


Weight. 


Country. 

U.S.  Gals. 

Country. 

U.  8.  Gals' 

England,  .  . 
France,  .  .  . 

..1  Gallon.     —  1.2 
.  .1  Dekaliter.  =2.64 

Switzerland, 
Turkey,  

1  Pot,          -    .40 
1  Almud,    —  1.38 

Prussia,  .  ,. 
Austria,  .  .. 

..1  Quart,      —    .30 
..IMaas,       -=    .37 

Mexico,  
Brazil,  

1  Fasco.     =    .63 
1  Medida.  =    .74 

Sweden.   .  . 

.  .  1  Kanna,     «=     .09 

Cuba  

1  Arroba.  =  4  01 

Denmark,  . 

.  .1  Kande,    =    .51 

South  Spain, 

1  Arroba.   =  4.25 

COMPARISON  OF 

GRAIN  MEASURES. 

Country. 

U.  S.  Bushels. 

Country 

U.  S.  Busheli. 

England,  .  . 
France,  ... 

..1  Bushel.  —  1.031 
.  .1  Hectoliter  =2.84 

Germany,  .  .  . 
Persia  

1  Schef.=  1.5to3 
1  Artaba.    —  1.85 

Prussia,  .  .. 

..1  Scheffel.—  1.56 

Turkey,  

1  Kilo,        =1.03 

Austria,    .  . 

..IMetze.     <=  1.75 

Brazil  

1  Fan.         =-  1.5 

Russia 

..IChetverik—    .74 

Mexico,    

1  Alque.     —  1.13 

Greece,  .  .  . 

.  .1  Kailou,    -2.837 

Madras,  

.iParah.      =1.743 

RAILROAD  FREIGHT. 


107 


COMPARATIVE  TABLE  OP  POUNDS  IN  DIFFERENT  COUNTRIES 


Austria,  100  Ibs 123.50  U.  S. 


Nedcrland,  100  Ibs . .  108.93  U. 


Bavaria,  kk 
Belgium,  u 
Bremen,  " 
Berlin,  " 
Denmark,  kk 
Ger.  Zoll.  States,  . 
Hamburg, 


.123.50 
,103,35 
.110.12 
.103.11 
.110.00 
.110.25 
.110.04 


Portugal,         kk 
Prussia, 
Russia,  kk 

Spain,  kk 

St.  Domingo,  kk 
Trieste,  kk 


..101.19 
..110.25 
..  90.00 
..101.44 
..107.93 
..123.60 


COMPARISON  OF  COMMERCIAL  WEIGHTS. 


Country. 

Austria, — 

Arabia, 

Brazil 

China. 

Denmark.  . . 
East  Indies. 

Egypt,   

France 

Germany,  . . 


•Weight-  U.  8.  Lbs. 

.1  Pfund.  —  1.23 
.1  Mannd.  —  .3 
.1  Arratel.— 
.1  Catty.  = 
.IPund.  — 
.1  Seer.  = 


1.02 
1.33 
1.10 
2.06 


.1  Rottoli.  =  1.0 
.1  Kilogram.  =2.20 
.1  Pfund,   =    1.10 


Country. 

Mexico, 
Madras,  .. 
Persia,  . 
Russia, 
Sweden, 
Spain.    . 
Sicily.    . 
Turkey.  . . 
Japan,  — 


Weight.  U.  B.  Lb«~ 

...1  Libra,  —    l.Ofr 

...IVis.  =3.125 

...1  Rattcl.  —  2.116 

...1  Funt,  —      .90- 

...IPund.  —      .93 

...1  Libra,  =>  1.016 

...1       -k  -      .7 

...1  Oka,  =    2.82 

...IKin,  =      .6$ 


Troy.  Apothecaries.  Avoirdupois. 

1  Pound  =  57GO  grains  =  57GO  grains  =  7000  grains. 
1  Ounce  =480       k>       =    480       "       =    437.5     kk 


1  Ounce  =480       k>       =    480       >%       =    437.5    kk 

175Pounds=   lt5  pounds^   144  pounds. 


RAILROAD  FREIGHT. — TABLE  OF  GROSS  WEIGHTS. 

When  the  actual  weights  are  not  known,  the  articles  are  billed 
as  per  the  following  table. 


Ale  and  Beer, 

320  Ib. 

per  bbl. 

Lime,  200  Ib. 

]>er  bbl» 

170    ;> 

"  1A  i% 

Malt,  38  k; 

kk     bu. 

H            tt                tt 

100    " 

"  J4  " 

Millet,  45  kk 

Apples,  dried, 
"        green, 

24  " 
50  lk 

"      bn 

Nails,  ..108  kk 
Oil,  400  kk 

kl  kegfc 
*;    bbl. 

Beef    ... 

150  fc* 
320  u 

kk     bbl. 

Peaches,  dried,    33  kk 
Pork                    320  " 

kk     bu. 
"    bbl 

Bran,  

..20    'l 

"      bn. 

Potatoes  (com  )  150  k; 

Brooms,  

..40  u 

ki    dox 

Salt    Fine           300  k* 

.1            U 

Cider,  

.350  " 

kk    bbl 

kk  Coarse          350  kk 

it              4t 

Charcoal,  

.    22  ik 

ik     bu 

kt  in  Sack        200  ki 

,      kk              It 

Eggs,  

.  200  lv 

kk    bbl. 

Turnips  56  kk 

kk     bu. 

Fish,  

.300  ** 

Vinegar              350  kk 

kk    bbl 

Flour,  

..200  " 

U 

Whiskey,  350  " 

Highwines,  .  .  . 

..350  tk 

U            U 

One  Ton  Weight,  

.20001b. 

108 


LIQUID,  OR  WINE,  MEASURE. 


CU.PT. 

CU.  IN. 

CU.FT. 

CU.  IN. 

.0167 

28.875 

4  Gills,..  ?  Pint. 

n.ae 

i9404 

2Tiecs..lPunsh'n. 

.0334 

57.75 

2  Pints..  1  Quart. 

4.2109 

7276.5 

31  Vi  Gals  IBbl. 

.1331 

231 

4  Qts.,  .  .  1  Gallon. 

8.421 

14553 

2Bbls  1  Hhd. 

1.331 

2310 

10  Gals...  1  Anker. 

16.84 

29106 

2  Hhds  1  Pipe. 

2.406 

4158 

18  Gals...  1  Runlet. 

33.68 

58212 

2  Pipes  1  Tun. 

5.614 

9702 

42  Gals.  .  .  1  Tierce. 

The  U.  S.  Standard  Gallon  contains  231  cubic  in.— 8^  Ibs.  avoird tip's. 
"    Imperial  ik  •*         277.274    "     =1.2  U.  S.  gallons. 

"    old  Beer  Measure  "  "        282          " 

In  measuring  tanks,  reservoirs,  etc.,  it  will  be  sufficiently  accu- 
rate to  regard  one  cubic  foot=7H  U.  S.  or  6J4  Imperial  gallons. 

The  contents  of  a  circular  tank,  in  barrels  of  31  yz  gallons,«=the 
square  of  the  diameter  (in  ft.)  multiplied  by  the  depth,  mul.  by  .1865. 


DRY  MEASURE,  U.  S.  STANDARD, 

For  measuring  Grain,  Fruit,  Roots,  Coal,  etc. 


CU.  FT. 

CU.  IN. 

PT. 

QT. 

GAL. 

PK. 

BU. 

CM. 

QR. 

.01944 

33.60 

....1 

1 

Pint. 

.03888 

67.20 

2 

....1 

1 

Quart. 

.1555 

268.80 

8 

4 

....1 

1 

Gallon. 

.3111 

537.60 

16 

8 

2 

1 

1 

Peck. 

1.2444 

2150.42 

64 

32 

8 

4 

1 

1 

Bushel. 

4.9778 

8601.68 

256 

128 

32 

16 

4 

..1 

..1 

Coomb. 

9.9556 

17203.36 

512 

256 

64 

32 

8 

2 

..1 

..1 

Quarter. 

39.8225 

68813.44 

2048 

1024 

256 

128 

32 

8 

4 

..1 

Chaldron. 

44.8004 

77415.12 

2304 

1152 

288 

144 

36 

OF   C 

OAL 

..1 

Chaldron. 

The  U.  S.  Standard  Bushel  contains  2150.42  cubic  inches. 
The  Imperial  English  "  "        2218.192     "    '   " 

A  cylinder  18^  inches  in  diameter,  8  inches  deep=  1  BusheL 
5  Stricken  measures=  4  heap  measures. 


POUNDS  IN  A  BUSHEL. 


109 


U.  S.  Bushels  X  1.03152;  the  product  will  be  Imperial  Bushels. 

Imperial  Bushels  -i-  1.03152;  the  quotient  will  be  U.  S.  Bushels. 

Any  three  factors  that  will  produce  the  number  of  inches  in  a 
given  quantity,  will  be  the  inside  dimensions  of  a  box  to  hold  that 
quantity ;  hence  a  box  11.2x16x12  in.,  will  contain  1  Standard  Bushel. 
924  cu.  inches  =  4  Liquid  Gallons ;  therefore  a  box  12X7X11  inches  will 
contain  4  gallons. 

An  open  box  made  with  the  greatest  economy  of  material ;  the  al- 
titude =  the  radius  of  the  Base ;  if  with  a  cover  the  altitude=  the  base. 

A  cubic  foot  =  8-10  of  a  bushel,  nearly ;  add  .44  of  a  bushel  for 
each  100  bushels. 

The  number  of  bushels  -j-  *£  =  the  number  of  cubic  feet. 

The  number  of  cubic  feet — l-5=the  number  of  Bushels. 

TABLE  OF  AVOIRDUPOIS  POUNDS  IN  A  BUSHEL, 

As  prescribed  by  statute  in  the  several  States  named. 


Commodities. 

I 

cj 

1 

4 

•d 

=5 

3' 

£ 

cs 
t-q 

S 

% 

S 

•sg 
•3 

C3 
C3 

^ 

1 

S 
fe; 

>H" 
^ 

0 

£ 
o 

C? 
C5 

$ 

^ 
OS* 

E^ 
£  fet 

Barley, 

50 

48 

48 

48 

48 

32 

46 

48 

48 

48 

48 

48 

48 

46 

47 

4645 

Beans, 

60 

60 

60 

GO 

60 

62 

Blue  SrassS'd 

14 

14 

14 

14 

14 

Buckwheat, 

40 

45 

40 

50 

52 

52 

46 

42 

42 

52 

50 

48 

42 

48 

4642 

Castor  Beans, 

46 

46 

46 

46 

Clover  Seed, 

60 

60 

60 

GO 

60 

60 

60 

64 

60 

60 

60 

60 

Dried  Apples, 

24 

25 

24 

28 

28 

24 

28 

28 

Dried  Peaches, 

33 

33 

33 

28 

28 

33 

28 

28 

Flax  Seed, 

56 

56 

56 

56 

56 

55 

55 

56 

Eemp  Seed, 

44 

44 

44 

44 

44 

Indian  Corn, 

52 

56 

52 

56 

56 

56 

56 

56 

56 

56 

52 

56 

58 

56 

56 

56 

56  56 

Corn,  in  ear, 

70 

68 

68 

Corn  Meal, 

48 

50 

50 

50 

50 

50 

Oats, 

32 

28 

32 

32 

35 

33^ 

32 

30 

30 

32 

32 

35 

30 

52 

32 

54 

32 

3236 

Onions, 

57 

48 

57 

57 

52 

57 

50 

50 

Potatoes, 

60 

60 

60 

60 

GO 

60 

60 

60 

60 

60 

60 

GO  GO 

Rye, 

,54 

56 

.54 

56 

56 

56 

32 

56 

56 

56 

56 

56 

56 

56 

56 

~>G 

5G  56 

Rye  Meal, 

.50 

50 

50 

Salt, 

50 

50 

50 

50 

->G 

Timothy  Seed, 

45 

45 

45 

45 

45 

44 

Wheat, 

60 

56 

60 

60 

60 

GO 

60 

60 

60 

GO 

60 

60 

60 

GO 

60 

60 

6060 

Wheat  Bran. 

20 

20 

20 

20 

The  price  per  cental  =  the  price  per  bushel  X  100-7-the  number  of  pounds 
in.  the  bushel.    See  page  45. 


110 


MEASURE  OF  TIME  TABLE. 


SEC. 

MIX. 

HRS. 

DA. 

WK. 

60 

1 

1 

Minute 

3GOO 

60 

....1 

1 

Hour 

86400 

1440 

24 

....1 

..1 

Day, 

604800 

10080 

168 

7 

..1 

..1 

Week, 

31536000 
31622400 

525600 
527040 

8760 
8784 

365 
366 

52 

..1 
..1 

Common  Year, 
Leap  Year. 

12  Calender  months  =  13  lunar  months  =  1  year. 

365  days,  5  hrs.  48  minutes,  50  seconds  =  1  Solar  year. 

10  years  =  1  decade.    10  decades  =  1  century. 

400  years  =  146,097  days,  a  number  exactly  divisible  by  7. 

The  civil  day  begins  and  ends  at  12  o'clock.  Midnight. 

The  Astronomical  day  begins  and  ends  at  12  o'clock,  Noon. 

As  the  year  contains  365J4  days,  nearly,  we  reckon  three  years  in 
every  four  as  containing  365  days,  and  the  fourth,  leap  year,  as  con- 
taining 366  days;  the  leap  year  is  always  a  multiple  of  4. 

The  even  centuries  not  divisable  by  400  are  not  leap  years. 

Formerly  the  new  year  began  on  the  25th  of  March  and  was  so 
reckoned  in  England  until  1753. 

In  ordinary  business  computations,  1  year  =  12  mos.  =  360  ds. 
1  month  =  30  days. 

-fl-2-fl  4-1  -f-l-fl  +1  +1 

Jan~  Feby.  Mar.  Apl.  May,  June,  July,  Aug.  Sept.  Oct.  Nov.  Dec. 
In  the  common  year  February  has  two  days  less  than  30,  in  leap 
year  1  day  less ;  seven  months  have  one  day  more. 

To  find  the  exact  number  of  days  between  two  dates. 

Multiply  the  number  of  entire  months  by  3,  call  the  product  tens; 
add  the  extra  days,  and  1  day  for  each  month  of  31  days ;  when  Feb'y 
occurs,  deduct  2  days  for  the  common,  and  1  day  for  Leap  year. 

How  many  days  from  1st  of  the  4th  month  to  9th  of  the  llth  month. 

11  mo.— 4  mo.  =  7  mo.       7x30+9+4=223  days. 


GOLD  AND  SILVER.  Ill 

DIAMOND  WEIGHT.  ASSAYERS'  WEIGHT. 

16  Parts  =  1  Grain.  240  Grains  =  1  Carat. 

4  Grains  =  1  Carat.  2  Carats      =  1  Ounce. 

1  Carat    =  31-5  Troy  grs.  (nearly.)    24  Carats    =  1  Pound. 
The  term  Carat  is  also  employed  in  estimating  the  fineness  of 
Gold  and  Silver;  when  perfectly  pure  the  metal  is  said  to  be  "24  Car- 
ats fine."  English  Gold  coin  is  22  carats  fine,  that  is,  it  consists  of 
22-24  pure  gold,  and  2-24  alloy. 

To  compute  the  fineness  in  thousandths,  and  the  weight  in  ounces 
and  thousandths  is  simpler,  and  admits  of  very  minute  subdivisions 
with  great  facility. 

The  coining  of  gold  or  silver  docs  not  change  the  REAL  value  of 
either;  it  stamps  each  piece  of  metal  with  a  national,  official  certificate 
of  its  weight  and  fineness. 

From  one  Troy  pound  of  gold  22  carats,  or  .916  2-3  fine  46  29-40 
>  Sovereigns  are  made,  each  weighing  123.27448  grains  =  113.001605 
grains  of  fine  gold  =  84.866563. 

1  ounce  of  U.  S.  Standard  Gold  =  $18.60465  =  £3.8230  =  £3,,16,,  5H 
1      kk       ^  British      "  k'     =    18.94918  =    3.8938  =    3,,17,,10^ 

1      "  -    "  Pure  k>     =    20.67184  =    4.248    =    4,,  4,,llfc 

Thousandths  of  an  ounce  -j-  100  X  48  =  grains. 
Grains  X  100  -f-  48  =  thousandths  of  an  ounce. 
U.  S.  Standard  ounces  of  Gold  -j-  .05375  =  U.  S.  Dollars. 
U.  S.  Gold  Dollars  X  .05375  =  Standard  ounces. 
To  multiply  by  .05375,  remove  the  point  one  place  to  the  left  and 
divide  by  2,  divide  this  quotient  by  20,  and  the  second  quotient  by  2; 
the  sum  of  the  quotients  is  the  answer. 

EXAMPLE.  —  How  many  ounces  in  one  U.  S.  Gold  dollar? 

A  _ 
20  .05 
.0025 

Alls.  .  05375  ozs. 
.  Oo375 

The  weight  of  gold,  in  ounces,  and  the  fineness  being  given,  to  find  its  val- 
ue in  U.  S.  Gold  Coin. 

RULE.  —  Multiply  the  weight  by  twice  the  fineness,  multiply  by  10 
and  divide  the  product  by  30,  and  the  quotient  by  129;  the  sum  of  the 
product  and  the  quotients  is  the  answer. 

EXAMPLE.  —  Find  the  value  of  one  ounce  of  gold  9-10  fine. 
3018. 


129 


.6 

_  .00465 

18760465  Ans.  $18.60465. 


112  HOWARD'S  ART  OF  COMPUTATION. 

Or  multiply  the  given  weight  by  the  fineness  X  1000  X  8,  and  divide 
the  product  by  387. 

IX.  9X1000X8-7-387  =  18.60465. 

The  fineness  and  weight  of  Silver  being  given,  to  Snd  its  value  in  U.  S. 
Silver  dollars  9-10  fine,  4i2yz  grains  weight. 

RULE. — For  pure  silver,  if  in  grains,  divide  by  9x10x11x3  and 
multiply  by  8,  or  divide  by  .9x412.5. 

EXAMPLE.— Pure  silver,  grains  371.25x8-j-9XlOxllX3=  $1. 
If  in  ounces,  divide  the  weight  and  fineness  by  .9  X  .895375. 
Or  multiply  the  given  weight  by  the  fineness  and  by  1.28;  repeat 
the  figures  in  the  product,  under,  and  two  places  to  the  right,  as  often, 
and  to  as  many  decimal  places  as  the  answer  requires  ;  the  sum  is 
the  answer. 

EXAMPLE. — Find  the  value  in  silver  dollars  of  1  oz.  of  silver  9-10  fine. 
1X.9X1.38  =  1.152 
1152 
1152 

$1.1636352  Ans. 

To  make  a  compound  of  any  weight  and  fineness. 

RULE.  Divide  the  fineness  sought  by  the  fineness  to  be  alloyed; 
the  quotient  is  the  weight  required  to  make  a  compound  of  one  ounce 
of  the  desired  fineness. 

EXAMPLE. — Required  to  make  a  compound  of  one  ounce  14  carats 
fine  by  alloying  gold  22  carats  fine. 

14-4-22  =  .63636  gold  +  .36364  alloy  =  1  ounce. 

To  find  how  many  ounces  of  a  lower  fineness  must  be  added  to  one  ounce 
of  a  higher  fineness  to  make  a  compound  of  any  given  fineness. 

RULE. — Divide  the  difference  of  the  two  higher  by  the  difference 
of  the  two  lower  finenesses. 

EXAMPLE.— Required  a  compound  of  14  Carats  fine  by  mixing  12 
carat  fine  with  21  carat  fine. 

14 zl.  =  3%.  3l/t  oz.  12  fine  +  1  oz.  21  fine  =  4J4  oz.  14  Carat  fine. 
14  — 12—2 

The  silver  dollar  weighs  412V4  grains,  nine-tenths  of  which  is 
pure  silver.  At  the  English  mint,  a  mixture  of  11  ozs.,  2  pwts.  of 
pure  silver,  with  18  pwts.  of  |alloy,  is  coined  into  66  shillings. 
When  English  coin  silver  is  worth  54  pence  an  ounce,  in  gold,  and  the 
pound  stg.  .(gold)  is  worth  $4,86  in  United  States  gold,  what  is  the 
value  in  U.  S.  gold  coin  of  the  silver  contained  in  the  dollar  ?  (The 
value  of  the  alloy  in  the  English  silver  is  not  to  be  considered. 

222 
11  ozs.,  2  pwts.  =  o7Q-=  -925  of  an  ounce*  Ans-  89^  cts< 

54  pence  =  £0.225.      .225X4.86  =  1.0935.        1.0935  X  412.5  X  .9  __ 

480  X  .925 


GOLD  AND  SILVER. 


113 


Estimate,  in  Millions,  from  the  latest  official  data,  of  the  Popu- 
lation, Imports  and  Exports,  National  Debts,  and  present  stock  of 
Gold  and  Silver  Coin  and  Bullion  in  the  world,  in  U.  S.  dollars  : 


Country. 

Pop. 

Impts 

Expts 

Debt. 

Gold. 

Silver. 

United  States,  

45 

466 

739 

2256 

245 

85 

Other  American  States,  .  .  . 
France,  

40 
37 

243 

892 

268 
961 

1250 
3750 

50 
1300 

50 
350 

Great  Britain  and  Colonies 
Germany,  

40 
40 

2109 
918 

1397 

608 

4308 
86 

650 
225 

100 

175 

Other  European  States,  .  .  . 
China,  

200 
400 

1790 
105 

1429 
114 

9418 
11 

300 
50 

300 

800 

British  India,  

240 

244 

325 

694 

100 

500 

Japan     

33 

24 

27 

349 

40 

10 

Other  Asiatic  States,  

65 

45 

75 

50 

200 

Africa  and  the  Islands,  .  .  . 

65 

85 

100 

450 

50 

20 

Total, 

1,205 

6,921 

6,043 

22,572 

3,060 

2,590 

Municipal  and  other  public  debts  are  not  here  included.  The 
city  of  Paris  owes  $459,000,000  ;  United  States  cities,  $550,000,000. 

The  quantity  of  Gold  and  Silver  in  the  form  of  Plate  is  perhaps 
equal  to  that  in  the  form  of  Coin  and  Bullion. 

The  product  of  the  Gold  and  Silver  mines  of  the  world  last  year 
was  about  $170,000,000  ;  the  mines  of  the  United  States  furnished  : 
Gold,  $47,000,000;  Silver,  $46,000,000. 

APPROXIMATE  VALUE  OF  VARIOUS  METALS, 

PER  POUND  AVOIRDUPOIS. 

Indium, $2522 

Vanadium 2510 

Ruthenium, 1400 

Rhodium 700 

Palladium,  653 

Uranium 576 

Titanium, 500 

Osmium 325 

Iridium, 317.44 

Gold   301.46 

Platinum 115.20 

Thallium 108.77 

Chromium 58.00 

Magnesium, 46.50 

Potassium 23.00 


£518     4     9 

Silver 

$1885 

£3  17  ,  6 

515,.  15,,  5 
287,,13,,  7 
143.,16,,10 
134;,  3,,  8 
118.,  7,,  3 

Cobalt, 
Cadmium,  . 
Bismuth,  .. 
Sodium,   .  .  . 
Nickel,  .... 

...  7.75 

...  6.00 
...  3.63 
.  ..  3.20 
...  2.50 

l"  4"  8 
0,,15,,  0 
0,,13,,  0 
0,,10,,  3 

Mercury,  .  .  . 

...   1.35 

0,,  5,,  6 

66,.15,,  9 
1     65,.  4,,  6 

Antimony.  . 
Tin,  

...     .36 
...     .33 

0,,  1,,  6 
0,,  1,,  4 

3     61  .18,,  11 

Copper 

...     .25 

0,,  1.,  0 

)     23.,13,,  5 
1     22.,  7,,  0 
}     11.,  18,,  3 
)       9..11,, 
3       4,,14,.  6 

Arsenic,  ... 
Zinc  
Lead,  
Iron,  

...     .15 
...     .11 
...     .07 
...     .02 

0,,  0,,  7 
0,,  0,,  5 
0,,  0,,  3 
0,,  0,,  1 

114        HOWARD'S  ART  OF  COMPUTATION. 

MISCELLANEOUS. 

How  many  strokes  does  a  clock  strike  in  12  hours  ? 

"12+1X12    7Q    ,    , 
-  n  -  —To  strokes. 
A 

How  many  barrels  in  a  triangular  pile,  49  bar- 
rels at  the  base  and  1  at  the  top? 


O'Leary  with  ten  tramps  have  two  days  start, 
and  make  8  miles  a  day  ;  how  long  will  it  take  Row- 
ell  with  5  trampers  travelling  10  miles  a  day  to 
overtake  O'Leary  and  his  men? 
16—2=8  days. 

The  sum  of  two  numbers  is  140  ;  the  larger  is  to 
the  smaller  as  1  to  9,  what  are  the  numbers  ? 


9     5     14 

99     9          140x  i54  =50  | 

A  Bin  9  ft.  6  in.  long,  6  ft.  wide,  4  ft.  3  in.  deep, 
will  hold  how  many  Imperial  bushels. 

•¥-XfX-Y-XT8<T—  4.845=  188.  955  bushels.  Ans. 

NOTE.  The  imperial  bushel  is  2218.192  Inches,  ten  eighths 
of  a  foot,nearly,  deduct  2£  from  every  100  bushels  in  the  product, 
this  result  multiplied  by  8  will  be  the  number  of  Imp.  gallons, 

What  is  the  cost  of  732  Ibs.  of  Coal  at  $14.  per  ton, 
2240  Ibs.  to  the  ton? 

732X14  , 

=$4'5'°-  Ans 


MISCELLANEOUS.  115 

A  bin  9  ft,  6  in.  long,  6  ft.  wide,  and  4  ft.  3  in. 
deep  is  full  of  wheat,  what  is  its  value  at  $2.05  a 
bushel  ? 

7.     Ans. 


Note.     The  standard  bushel   is  2150.42   inches;  ten-eighths 
of  a  foot,  nearly,  the  difference  is  .44  bu.  in  each  100.    #.259, 

Divide  £1  into  3  parts  in  the  proportion  of  A,  J, 
B,  i,  C,  J.         6+4+3=13. 

12  Ans.  A,  A,  fV 

How  many  cubic  feet  in  a  case  3  ft.  6  in.  by  2  fu 
8  in.  by  1  ft.  10  in? 

JXSX¥=17ift.     Ans. 

If  7  cats,  kill  7  rats,  in  7  minutes,  how  many  cats 
will  kill  100  rats  in  50  minutes? 
7X7X100=14. 

7X50  Ans.  14  cats. 

If  it  cost  $24  to  carry  6  tons  20  miles,   what  will 
it  cost  to  carry  12  tons  120  miles? 
24X12X120=288. 

6X20  Ans.  $288. 

How  many  bricks  will  pave  a  walk  200   ft.    long, 
by  16  feet;  bricks  8  in.,  by  4  in? 
200X16X3X3 

2X1"   14>400'  Ans.  14400  bricks. 

Multiply  £19  19s.  11  f  d  by  19i*  ¥V*  *»„. 


£399  J§  Tjf 

or*19,,19,,llf  X  20  -  ^  =  £899,,19,,2  ^  of  a 
farthing. 


116         HOWARD'S  ART  OF  COMPUTATION. 

Multiply  66  by  f  :    22  -^^      =     44. 
Divide  66  by  f:    33 


Divide  IG8x2x7\)y7x3: 

/  ^  r 


Divide  £99  amongst  3  persons,  A  to  have  T\,  B  T4T, 
and  C  T\. 


, 

Two  merchants  load  a  ship  with  goods  worth  £5000, 
A  owns  .£3500,  and  B  the  rest  ;  the  goods  suffer 
damage  valued  at  XI  000,  what  is  each  man's  share  o£ 
the  loss  ? 

1WP        wm     1PW        A  loses  .£700. 
3500  r     1500         B      „      £300. 

B  and  C  gain  by  trade  £182;  B  put  in  £300,  and 
C  £400,  what  is  the  gain  of  each  ? 

*<m  I  3(^  >™  I  W  B  £78. 

'W  I  182  W  I  182  C  £104. 

A  person  owning  f  of  a  mine  sells  |  of  his  share  for 
X1710,  what  is  the  value  of  the  whole  mine  ? 


190  -___=    ^3800. 

f    X    p 

How  much  money  will  buv  |  of  f  of  a  mine  worth 
£3800  ? 


1X1  =  A  -      -     £1710. 

If  J  of  6  be  3,  what  will  J  of  20  be  ? 
3  X  ? 

2   (3  X 


MISCELLANEOUS.  117 

A  compositor  can  set  20  pages  in  f  of  a  day,  an- 
other could  set  20  pages  in  f  of  a  day,  how  long 
'•will  it  take  the  two  men  working  together  to  do  the 
ivork? 

4     5     23  23  6 

-H  —  =  —  —  inverted   =  —  of  a  day. 

326  6  23 

A  cistern  has  5  faucets  ;  the  first  will  fill  it  in  1 
hour,  the  second  in  two,  the  third  in  3,  the  fourth  in 
4,  and  the  fifth  in  5  hours  ;  in  what  time  will  the  cis- 
tern be  filled,  all  the  faucets  running  at  once  ? 
60+30+20+15+12    137  137.          60 


A  says  to  B,  give  me  $7  and  I  shall  have  as  much 
money  as  you  ;  B  replies,  give  me  $7  and  I  shall  have 
twice  as  much  as  you  ;  how  much  money  had  each  ? 
7x5-35  7X7=49  A  $35,  B  $49. 

How  many  different  pairs  can  be  made  with  7 
units  ? 

—  ^  —  =  21  pairs. 

tU 

How  many  bricks,  8x4x2  inches,  in  a  wall 
160x20x2  feet? 

160X20X2X3X3X6  ^ 

2xlXl 

How  many  shingles  for  a  roof  60  ft.  long,  rafters 
20  feet,  two  sides,  shingles  to  show  6x4  inches.1 
60X20x2x2x8=14>400 

J.  X  -*- 


118         HOWARD'S  ART  OF  COMPUTATION. 

If  21  f  bushels  of  oats  will  seed  9^  acres,  how 
many  bushels  will  seed  100  acres? 
.87x3x100  •      • 

4x29     =225  bushels- 

How  many  16ths  are  there  in  .85  ? 
.85X16         __ 

^oo- 

$150  is  due  Jan.  1st.,  $78  is  paid  down,  on  July 
1st.,  the  account  is. settled  by  paying  $78.  What 
rate  per  cent  is  paid  for  the  accomodation  ? 

fc79  6X2X100_1 
72 

Find  the  value  of  an  ounce  of  silver,  gold  being 
worth  £3,,18,, 7  per  ounce,  ratio  15£to  1.  also  16  to  1. 
£3,,18,,7  -s-15£=60$$d.     £3,,18,,7-=-l6=58^d. 

What  is  the  interest  on  980  dollars  for  six  days 
at  7  per  cent,  per  annum  ? 
980 
98 
49 


$150— 78=472.    ^  ^ =165  l^r  cent. 


1.127  Ans.  SI.  127. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


OCT    ^3   JS36 

SFKfT  ON  ILL 

AMP      •»      A     ^(\t\"^% 

All  b  1  4  ?007 

U.U.  BERKELEY 

I  LJ      I  H  \ 


